Applications of Particle Physics to the Early
Universe
by
Leonardo Senatore
Submitted to the Department of Physics
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy in Physics
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2006
( Massachusetts Institute of Technology 2006. All rights reserved.
Author........................................-.
Department of Physics
May 5, 2006
Certified by.
.............................................
Alan Guth
Victor F. Weisskopf Professor of Physics
;ghegis Supqrvisor
Certifiedby................
V!im`n ArkaM-Hamed
Professor of Physics
Thesis Supervisor
Acceptedby...............
.......... V-.' ' .f. ..'. . ""'Thoma
' Greytak
Associate Department Head ff Education
MASSACHU
i i $IIThT
NO
OF TECHNOLOGY
Al lt.
nr
Lzz-%00I
uu
.
LIBRARIES
ARCHIVES
Applications of Particle Physics to the Early Universe
by
Leonardo Senatore
Submitted to the Department of Physics
on May 5, 2006, in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy in Physics
Abstract
In this thesis, I show some of the results of my research work in the field at the
crossing between Cosmology and Particle Physics. The Cosmology of several models
of the Physics Beyond the Standard Model is studied. These range from an inflationary model based on the condensation of a ghost-like scalar field, to several models
motivated by the possibility that our theory is described by a landscape of vacua, as
probably implied by String Theory, which have influence on the theory of Baryogenesis, of Dark Matter, and of Big Bang Nucleosynthesis. The analysis of the data of
the experiment WMAP on the CMB for the search of a non-Gaussian signal is also
presented and it results in an upper limit on the amount on non-Gaussianities which
is at present the most precise and extended available.
Thesis Supervisor: Alan Guth
Title: Victor F. Weisskopf Professor of Physics
Thesis Supervisor: Nima Arkani-Hamed
Title: Professor of Physics
2
Acknowledgments
I would like to acknowledge my supervisors, Prof. Alan Guth, from whom I learnt
the scientific rigor, and Prof. Nima Arkani-Hamed, whose enthusiasm is always an
inspiration; the other professors with whom I had the pleasure to interact with: Prof.
Matias Zaldarriaga, Prof. Max Tegmark, and in particular Prof. Paolo Creminelli,
with whom I ended up spending a large fraction of my time both at work and out
of work; all the many Post-Docs and students who helped me, in particular Alberto
Nicolis; and eventually all the friends I have met in the journey which brought me
here. Often I was in the need of help, and I have always found someone that, smiling,
was happy to give it to me. It is difficult to imagine, being a foreign far from home,
if without all these people I would have been able to arrive to this point.
At the end, a special thank must go to my parents and to my girlfriend, who have
always been close to me.
3
Contents
1 Introduction
7
2 Tilted Ghost Inflation
14
2.1
Introduction.
. . . . . . . . . . . . . . . . . . . .
14
2.2
Density Perturbations.
. . . . . . . . . . . . . . . . . . . .
16
. . . . . . . . . . . . . . . . . . . .
18
2.3 Negative Tilt.
2.3.1
2-Point Function
.....
. . . . . . . . . . . . . . . . . . . .
18
2.3.2
3-Point Function
.....
. . . . . . . . . . . . . . . . . . . .
21
2.3.3
Observational Constraints
. . . . . . . . . . . . . . . . . . . .
27
2.4 Positive Tilt ............
. . . . . . . . . . . . . . . . . . . .
29
2.5
. . . . . . . . . . . . . . . . . . . .
33
Conclusions ............
3 How heavy can the Fermions in Spliit Susy be? A study on Gravitino
and Extradimensional LSP.
36
3.1
Introduction.
. . . . . . . . . . . . . . . . . . . .
37
3.2
Gauge Coupling Unification . . . . . . . . . . . . . . . . . . . . . . .
39
3.2.1
Gaugino Mass Unification
42
3.2.2
Gaugino Mass Condition froom Anomaly Mediation
. . . . . . . . . . . . . . . . . . . .
......
45
3.3
Hidden sector LSP and Dark Matt,er Abundance
3.4
Cosmological Constraints .....
. . . . . . . . . . . . . . . . . . . .
51
3.5
Gravitino LSP ..........
. . . . . . . . . . . . . . . . . . . .
56
3.5.1
Neutral Higgsino, Neutral
AVino,
3.5.2 Photino NLSP .....
4
. . . . . . . . ....
48
and Chargino NLSP .....
58
. . . . . . . . . . . . . . . . . . . .
61
3.5.3
............. .. 63
Bino NLSP.
3.6
Extradimansional LSP .....
. . . . . . . . . . . . . .
64
3.7
Conclusions ...........
. . . . . . . . . . . . . .
68
3.7.1
Gravitino LSP ......
. . . . . . . . . . . . . .
69
3.7.2
ExtraDimensional LSP .
. . . . . . . . . . . . . .
69
4 Hierarchy from Baryogenesis
4.1
Introduction
4.2
Hierarchy
76
. . . . . . . . . . . . . . . .
from Baryogenesis
. . . . . . . . ......
. . . . . . . . . . . . . . .
4.2.1 Phase Transition more in detail
4.3
Electroweak Baryogenesis
.
. ....
80
..........
85
....................
4.3.1
CP violation sources ..............
4.3.2
Diffusion
Equations
.
77
89
91
. . . . . . . . . . . . . . .
......
4.4
Electric Dipole Moment
4.5
Comments on Dark Matter and Gauge Coupling Unification
4.6
Conclusions
4.7
Appendix: Baryogenesis in the Large Velocity Approximation
100
.........................
. . . . . . . . . . . . . . . .
106
.
.....
111
. . . . . . . ......
112
....
114
5 Limits on non-Gaussianities from WMAP data
123
5.1
Introduction.
.......
124
5.2
A factorizable equilateral shape ..............
.......
127
5.3
Optimal estimator for fNL
.......
133
5.4
Analysis of WMAP 1-year data
.......
137
5.5
Conclusions .........................
.......
144
.................
..............
6 The Minimal Model for Dark Matter and Unification
154
.......
.......
155
158
6.1
Introduction.
6.2
The Model .
6.3
Relic Abundance.
. . . .
159
..
6.3.1
....
.
161
..
....
.
163
..
Higgsino Dark Matter.
6.3.2 Bino Dark Matter .
5
Direct Detection
6.5
Electric Dipole Moment
6.6
Gauge Coupling
Unification
6.6.1
and matching
Running
165
.............................
6.4
.........................
166
. . . . . . . . . . . . . .
. . .
. . . . . . . . . . . . . . . . . ...
6.7
Conclusion
6.8
Appendix A: The neutralino mass matrix
................................
..
168
171
178
.
...............
6.9 Appendix B: Two-Loop Beta Functions for Gauge Couplings .....
7 Conclusions
179
180
187
6
Chapter
1
Introduction
During my studies towards a PhD in Theoretical Physics, I have focused mainly on
the connections between Cosmology and Particle Physics.
In the very last few years, there has been a huge improvement in the field of Observational Cosmology, in particular thanks to the experiments on the cosmic microwave
background (CMB) and on large scale structures. The much larger precision of the
observational data in comparison with just a few years ago has made it possible to
make the connection with the field of Theoretical Physics much stronger. This has
occurred at a time in which the field of Particle Physics is experiencing a deficiency
of experimental data, so that the indications coming from the observational field of
Cosmology have become very important, or even vital. This is particularly true, for
example, for the possible Physics associated with the ultra-violet (UV) completion of
Einstein's General Relativity (GR), as, for example, String Theory.
Because of this, Cosmology has become both a field where new ideas from Particle
Physics can be tested, potentially even verified or ruled out with great confidence,
and finally applied in search for a more complete understanding of the early history
of the universe, and also a field which, with its observations, can motivate new ideas
on the Physics of the fundamental interactions.
Some scientific events which occurred during my years as a graduate student have
particularly influenced my work.
From the cosmological side, the results produced by the Wilkinsons Microwave
7
Anisotropy Probe (WMAP) experiment on the CMB have greatly improved our
knowledge of the current cosmological parameters [1]. These new measurements have
gone into the direction of verifying the generic predictions of the presence of a primordial inflationary phase in our universe, even though not yet at the necessary high
confidence level to insert inflation in the standard paradigm for the history of the
early universe. The first careful study of the non-Gaussian features of the CMB has
been of particular importance for my work, as I will highlight later. Further, these
new data have shown with high confidence the detection of the presence of a form
of energy, called Dark Energy, very similar to the one associated with a cosmological
constant, which at present dominates the whole energy of the universe.
This last very unexpected discovery from Cosmology is an example of how cosmological observations can affect the field of Theoretical Physics. In fact, this observation
has been of great influence on the development of a new set of ideas in Theoretical
Physics which have strongly influenced my work. In fact, the problem of the smallness
of the cosmological constant is one of the biggest unsolved problems of present Theoretical Physics, and, until these discoveries, it had been hoped that a UV complete
theory of Quantum Gravity, once understood, would have given a solution. However,
from the theoretical side, String Theory, which is, up to now, the only viable candidate for being a UV completion of GR, has shown that there exist consistent solutions
with a non-null value of the cosmological constant, and that there could be a huge
Landscape of consistent solutions, with different low energy parameters. Therefore,
the theoretical possibility that the cosmological constant could be non zero, and that
it could have many different values according to the different possible histories of the
universe, together with the observational fact that the cosmological constant in our
universe appears to be non zero, has led to a revival of a proposal by Weinberg [2]
that the environmental, or anthropic, argument that structures should be present in
our universe in order for there to be life, could force the value of the cosmological
constant to be small. Recently, this argument has then been extended to the possibility that also other parameters of the low energy theory might be determined by
environmental arguments [3, 4]. For obvious reasons, this general class of ideas is
8
usually referred to as Landscape motivated, and here I will do the same.
In this thesis, I will illustrate some of the works I did into these directions. I consider useful to explain them in the chronological order in which they were developed.
In chapter 2, I will develop a model of inflation where the acceleration of the
universe is driven by the ghost condensate [5] . The ghost condensate is a recently
proposed mechanism through which the vacuum of a scalar field can be characterized
by a constant, not null, speed for the field [6]. The inflationary model based on this has
the interesting features of generically predicting a detectable level of non-Gaussianities
in the CMB, and of being the first example which shows that it is possible to have an
inflationary model where the Hubble rate grows with time. The studies performed in
this work have resulted in the publication in the journal Physical Review D of the
paper in [7].
In chapter 3, in the context of Split Supersymmetry [3], which is a recently proposed supersymmetric model motivated by the Landscape where the scalar superpartners are very heavy, and the higgs mass is finely tuned for environmental reasons,
I will study the possibility that the dark matter of the universe is made up of particles which interact only gravitationally, such as the gravitino. This possibility, if
realized in nature, would have deep consequences on the possible spectrum of Split
Supersymmetry and its possible detection at the next Large Hadron Collider (LHC).
I will conclude that observational bounds from Big Bang Nucleosynthesis strongly
constrain the possibility that the gravitino is the dark matter in the context of Split
Supersimmetry, while other hidden sector candidates are still allowed. The studies performed in this work have resulted in the publication in the journal Physical
Review D of the paper in [8].
In chapter 4, I will study a model developed in the context of the Landscape,
in which the hierarchy between the Weak scale and the Plank scale is explained in
terms of requiring that baryons are generated in our universe through the mechanism
of Electroweak Baryogenesis. While the signature of this model at LHC could be
9
very little, the constraint from baryogenesis implies that the system is expected to
show up within next generation experiments on the electron electric dipole moment
(EDM). The studies performed in this work have resulted in the publication in the
journal Physical Review D of the paper in [9].
In chapter 5, I will show a work in which I performed the analysis of the actual data
on the CMB from the WMAP experiment, in search of a signal of non-Gaussianity in
the temperature fluctuations. I will improve the analysis done by the WMAP team,
and I will extend it to other physically motivated models. At the moment, the limit
I will quote on the amplitude of the non-Gaussian signal is the most stringent and
complete available. The studies performed in this work have resulted in the publication of the paper in [10], accepted for publication by the Journal of Cosmology
and Astroparticle
Physics in May 2006.
In chapter 6, again inspired by the possibility that there is a Landscape of vacua
in the fundamental theory, and that the Higgs particle mass might be finely tuned for
environmental reasons, I will study the minimal model beyond the Standard Model
that would account for the dark matter in the universe as well as gauge coupling
unification. I will find that this minimal model can allow for exceptionally heavy
dark matter particles, well beyond the reach of LHC, but that the model should
however reveal itself within next generation experiments on the EDM. I will embed
the model in a Grand Unified Extradimenesional Theory with an extra dimension in
an interval, and I will study in detail its properties. The studies performed in this
work has resulted in the publication in the journal Physical Review D of the paper
in [11].
This will conclude a summary of some of the research work I have been doing
during these years.
Both the field of Cosmology and of Particle Physics are expecting great results
from the very next years.
Improvement of the data from WMAP experiment, as well as the launch of Plank
and CMBPol experiments are expected to give us enough data to understand with
10
confidence the primordial inflationary era of our universe. A detection in the CMB of
a tilt in the power spectrum of the scalar perturbations, or of a polarization induced
by a background of gravitational waves, or of a non-Gaussian signal would shed light
into the early phases of the universe, and therefore also on the Physics at very high
energy well beyond what at present is conceivable to reach at particle accelerators.
Combined with a definite improvement in the data on the Supernovae, expected in
the very near future, these new experiment might shed light also on the great mystery
of the present value of the Dark Energy, testing with great accuracy the possibility
that it is constituted by a cosmological constant.
The turning on of LHC will probably let us understand if the solution to the
hierarchy problem is to be solved in some natural way, as for example with TeV scale
Supersymmetry, or if there is evidence of a tuning which points towards the presence
of a Landscape.
It is a pleasure to realize that by the next very few years, many of these deep
questions might have an answer.
11
Bibliography
[1] D. N. Spergel et al. [WMAP Collaboration], "First Year Wilkinson Microwave
Anisotropy Probe (WMAP) Observations: Determination of Cosmological Parameters," Astrophys. J. Suppl. 148, 175 (2003) [arXiv:astro-ph/0302209].
[2] S. Weinberg, "Anthropic Bound On The Cosmological Constant," Phys. Rev.
Lett. 59 (1987) 2607.
[3] N. Arkani-Hamed and S. Dimopoulos, "Supersymmetric unification without low
energy supersymmetry and signatures for fine-tuning at the LHC," JHEP 0506,
073 (2005) [arXiv:hep-th/0405159].
[4] N. Arkani-Hamed, S. Dimopoulos and S. Kachru, "Predictive landscapes and
new physics at a TeV," arXiv:hep-th/0501082.
[5] N. Arkani-Hamed, P. Creminelli, S. Mukohyama and M. Zaldarriaga, "Ghost
inflation," JCAP 0404, 001 (2004) [arXiv:hep-th/0312100].
[6] N. Arkani-Hamed,
H. C. Cheng, M. A. Luty and S. Mukohyama,
"Ghost con-
densation and a consistent infrared modification of gravity," JHEP 0405, 074
(2004) [arXiv:hep-th/0312099].
[7] L. Senatore,
"Tilted
ghost
inflation,"
Phys.
Rev.
D 71,
043512
(2005)
[arXiv:astro-ph/0406187].
[8] L. Senatore, "How heavy can the fermions in split SUSY be? A study on gravitino and extradimensional LSP," Phys. Rev. D 71, 103510 (2005) [arXiv:hepph/0412103].
12
[9] L. Senatore, "Hierarchy from baryogenesis," Phys. Rev. D 73, 043513 (2006)
[arXiv:hep-ph/0507257].
[10] P. Creminelli, A. Nicolis, L. Senatore, M. Tegmark and M. Zaldarriaga,
"Limits
on non-Gaussianities from WMAP data," arXiv:astro-ph/0509029.
[11] R. Mahbubani and L. Senatore, "The minimal model for dark matter and unification," Phys. Rev. D 73, 043510 (2006) [arXiv:hep-ph/0510064].
13
Chapter 2
Tilted Ghost Inflation
In a ghost inflationary scenario, we study the observational consequences of a tilt in
the potential of the ghost condensate. We show how the presence of a tilt tends to
make contact between the natural predictions of ghost inflation and the ones of slow
roll inflation. In the case of positive tilt, we are able to build an inflationary model
in which the Hubble constant H is growing with time. We compute the amplitude
and the tilt of the 2-point function, as well as the 3-point function, for both the cases
of positive and negative tilt. We find that a good fraction of the parameter space of
the model is within experimental reach.
2.1
Introduction
Inflation is a very attractive paradigm for the early stage of the universe, being able
to solve the flatness, horizon, monopoles problems, and providing a mechanism to
generate the metric perturbations that we see today in the CMB [1].
Recently, ghost inflation has been proposed as a new way for producing an epoch
of inflation, through a mechanism different from that of slow roll inflation [2, 3]. It
can be thought of as arising from a derivatively coupled ghost scalar field
14
which
condenses in a background where it has a non zero velocity:
(X)=
2
(q) = M2 t
(2.1)
where we take M 2 to be positive.
Unlike other scalar fields, the velocity (X) does not redshift to zero as the universe
expands, but it stays constant, and indeed the energy momentum tensor is identical
of that of a cosmological constant. However, the ghost condensate is a physical fluid,
and so, it has physical fluctuations which can be defined as:
= M 2t + 7r
(2.2)
The ghost condensate then gives an alternative way of realizing De Sitter phases
in the universe. The symmetries of the theory allow us to construct a systematic
and reliable effective Lagrangian for 7r and gravity at energies lower than the ghost
cut-off M. Neglecting the interactions with gravity, around flat space, the effective
Lagrangian for r has the form:
a
S d4X*2
2
21~2
(V27r)2
2i *(V7r)2 +
2M 2
(2.3)
where a and 3 are order one coefficients. In [2], it was shown that, in order for the
ghost condensate to be able to implement inflation, the shift symmetry of the ghost
field p had to be broken. This could be realized adding a potential to the ghost.
The observational consequences of the theory were no tilt in the power spectrum, a
relevant amount of non gaussianities, and the absence of gravitational waves. The non
gaussianities appeared to be the aspect closest to a possible detection by experiments
such as WMAP. Also the shape of the 3-point function of the curvature perturbation
Cwas different from the one predicted in standard inflation. In the same paper [2], the
authors studied the possibility of adding a small tilt to the ghost potential, and they
did some order of magnitude estimate of the consequences in the case the potential
decreases while
X
increases.
15
In this chapter, we perform a more precise analysis of the observational consequences of a ghost inflation with a tilt in the potential. We study the 2-point and
3-point functions. In particular, we also imagine that the potential is tilted in such a
way that actually the potential increases as the value of
X
increases with time. This
configuration still allows inflation, since the main contribution to the motion of the
ghost comes from the condensation of the ghost, which is only slightly affected by the
presence of a small tilt in the potential. This provides an inflationary model in which
H is growing with time. We study the 2-point and 3-point function also in this case.
The chapter is organized as follows. In section 2.2, we introduce the concept
of the tilt in the ghost potential; in section 2.3 we study the case of negative tilt,
we compute the 2-point and 3-point functions, and we determine the region of the
parameter space which is not ruled out by observations; in section 2.4 we do the same
as we did in section 2.3 for the case of positive tilt; in section 2.5 we summarize our
conclusions.
2.2 Density Perturbations
In an inflationary scenario, we are interested in the quantum fluctuations of the r
field, which, out of the horizon, become classical fluctuations. In [3], it was shown
that, in the case of ghost inflation, in longitudinal gauge, the gravitational potential
1Idecays to zero out of the horizon. So, the Bardeen variable is simply:
H
-- H=
-
(2.4)
and is constant on superhorizon scales. It was also shown that the presence of a ghost
condensate modifies gravity on a time scale r
-1
, with F
M3 /Mpl, and on a length
scale m-l 1, with m - M 2 /Mpl. The fact that these two scales are different is not a
surprise since the ghost condensate breaks Lorentz symmetry.
Requiring that gravity is not modified today on scales smaller than the present
16
Hubble horizon, we have to impose
< Ho, which implies that gravity is not modified
during inflation:
r <<m < H
(2.5)
This is equivalent to the decoupling limit Mpl --+ o, keeping H fixed, which implies
that we can study the Lagrangian for r neglecting the metric perturbations.
Now, let us consider the case in which we have a tilt in the potential. Then, the
zero mode equation for r becomes:
+3H + V' = 0
(2.6)
which leads to the solution:
V'
r=
3H
(2.7)
We see that this is equivalent to changing the velocity of the ghost field.
In order for the effective field theory to be valid, we need that the velocity of 7rto
be much smaller than M 2 , so, in agreement with [2], we define the parameter:
62
V'
=-3HM 22 for V' < 0
3HM
V'
62 = +
for V' > 0
3HM 2
(2.8)
to be 62 < 1. We perform the analysis for small tilt, and so at first order in 62.
At this point, it is useful to write the 0-0 component of the stress energy tensor,
for the model of [3]:
To = -M 4P(X) + 2M4P'(X)q 2 + V(0)
(2.9)
where X = 09,Oq50u.
The authors show that the field, with no tilted potential, is
attracted to the minimum of the function P(X), such that, P(Xmi,) = M 2. So,
adding a tilt to the potential can be seen as shifting the ghost field away from the
minimum of P(X).
Now, we proceed to studying the two point function and the three point function
17
for both the cases of a positive tilt and a negative tilt.
2.3
Negative Tilt
Let us analyze the case V' < 0.
2.3.1
2-Point Function
To calculate the spectrum of the 7r fluctuations, we quantize the field as usual:
(2.10)
rk(t) = Wk(t)ak + W (t)at k
The dispersion relation for
Wk
is:
4
W2 = a-k + j2k2
k
M2
(2.11)
(2.11)
±
Note, as in [3], that the sign of p is the same as the sign of <
X
>= M 2 . In all this
chapter we shall restrict to P > 0, and so the sign of 3 is fixed.
We see that the tilt introduces a piece proportional to k2 in the dispersion relation.
This is a sign that the role of the tilt is to transform ghost inflation to the standard
slow roll inflation. In fact, w2 - k 2 is the usual dispersion relation for a light field.
Defining Wk(t)
=
uk(t)/a, and going to conformal time dr = dt/a, we get the
following equation of motion:
k4H22 272
up, + (/6 2 k2 + a kH
2
2)u k = 0
(2.12)
If we were able to solve this differential equation, than we could deduce the power
spectrum. But, unfortunately, we are not able to find an exact analytical solution.
Anyway, from (2.12), we can identify two regimes: one in which the term
dominates at freezing out, w - H, and one in which it is the term in
-
k4
k2 that
dominates at that time. Physically, we know that most of the contribution to the
18
shape of the wavefunction comes from the time around horizon crossing. So, in order
for the tilt to leave a signature on the wavefunction, we need it to dominate before
freezing out.There will be an intermediate regime in which both the terms in k2 and
k4 will be important around horizon crossing, but we decide not to analyze that case
as not too much relevant to our discussion. So, we restrict to:
62
>
a1/2 H
>2
(2.13)
_
where cr stays for crossing. In that case, the term in k 2 dominates before freezing
out, and we can approximate the differential equation (2.12) to:
2
"2
Uk+ (k2 -
(2.14)
)uk= 0
where k = 3 1/ 2 5k. Notice that this is the same differential equation we would get for
the slow roll inflation upon replacing k with k.
Solving with the usual vacuum initial condition, we get:
Wk =-H7
e-i0
/2
i
(1-
)
(2.15)
which leads to the power spectrum:
P=
and, using
=
k3
2 Iwk(
0)12
Wk (q2
=
4'
H2
2
/33 / 2 63
(2.16)
7T,
H4
PC = 47r2o3/253M 4
(2.17)
This is the same result as in slow roll inflation, replacing k with k. Notice that,
on the contrary with respect to standard slow roll inflation, the denominator is not
suppressed by slow roll parameters, but by the 62 term.
19
The tilt is given by:
2M 2V'
=
HV
dink
V"/
+ H2
3 dln
2 dlnk
dln(H)
dln(P¢)
dln(k)
1
2
262
9
3 dln52
2 dink
-
2 ddln) )Ik= aH =(2.18)
dink
p1/26
+ 4M4(1- 2P"M8))
where k = -r7/25is the momenta at freezing out, and where P and its derivatives are
evaluated at Xmin.
Notice the appearance of the term -
, which can easily be the dominant piece.
Please remind that, anyway, this is valid only for
62 >
6c,. Notice also that, for the
effective field theory to be valid, we need:
V'
(2.19)
3H
so, Mvi
<
M
. This last piece is in general << 1 if the ghost condensate is present
today. In order to get an estimate of the deviation from scale invariance, we can see
that the larger contribution comes from the piece in - vH2. From the validity of the
effective field theory, we get:
62 M 2 H
IV'I
IV"/A) = IV"I(M 2 /IH)Ne = V"l < 62
Ne
(2.20)
where Ne is the number of e-foldings to the end of inflation. So, we deduce that the
deviation of the tilt can be as large as:
Ins- -<N
Ne
This is a different prediction from the exact n8 = 1 in usual ghost inflation.
20
(2.21)
2.3.2
3-Point Function
Let us come to the computation of the three point function. The leading interaction
term (or the least irrelevant one), is given by [2]:
eHt
Lint = -,3 2M2 (*(V702)
(2.22)
Using the formula in [4]:
< ,k(t)l(t)7(t)
>=-
i
dt' < [rk,(t)7k2 (tk(t)l(t),
d3xHt(t')] >
(2.23)
we get [2]:
<7rkl7rk2 7rk3
>=
2 (27r35(
M2(2)6(
ki)
k)
(2.24)
wl(O)w2 (O)w3 (O)((k2 .k3 )I(1, 2, 3) + cyclic + c.c)
where cyclic stays for cyclic permutations of the k's, and where
f0
I(1,2,3) =
1
1
l()'w2* (v)w3(n)
(2.25)
and the integration is performed with the prescription that the oscillating functions
inside the horizon become exponentially decreasing as
--+-oo.
We can do the approximation of performing the integral with the wave function
(2.15). In fact, the typical behavior of the wavefunction will be to oscillate inside the
horizon, and to be constant outside of it. Since we are performing the integration on
a path which exponentially suppresses the wavefunction when it oscillates, and since
in the integrand there is a time derivative which suppresses the contribution when a
wavefunction is constant, we see that the main contribution to the three point function
comes from when the wavefunctions are around freezing out. Since, in that case, we
are guaranteed that the term in k2 dominates, then we can reliably approximate the
21
that ( = - H7r,weget:
wavefunctions in the integrand with those in (2.15). Using
H8 ~~~~~~
< (k(k
H8
3 6 sM
ki)4
2 ,k3 >= (27r)363(Z
(2.26)
3) ((k 2 + k3)kt + k2+ 2k 3 k2) + cyclic)
k3 i~=~lk (kl2(
where ki = ki . Let us define
F(k, k 2 , k3 ) =
3 3
which, apart for the
(k(k
2 .k3 ) ((k 2
+ k 3)kt + kt2+ 2k3k2) + cyclic)
(2.27)
function, holds the k dependence of the 3-point function.
The obtained result agrees with the order of magnitude estimates given in [2]:
<__3_>
(< (2 >)3/2
1
1
H
(< 1_(H)
¢3
>)I()/
8 M (1(H)2)3/2
6
el,
1 H
H
67/2 (M
-()2
(2.28)
(2
The total amount of non gaussianities is decreasing with the tilt. This is in agreement
with the fact that the tilt makes the ghost inflation model closer to slow roll inflation,
where, usually, the total amount of non gaussianities is too low to be detectable.
The 3-point function we obtained can be better understood if we do the following
observation. This function is made up of the sum of three terms, each one obtained
on cyclic permutations of the k's. Each of these terms can be split into a part which
is typical of the interaction and of scale invariance, and the rest which is due to the
wave function. For the first cyclic term, we have:
Interaction
(k 2 .k 3 )
(2.29)
while, the rest, which I will call wave function, is:
((k + k3 )kt + k + 2k 2 k 3)
2
Wavefunction =(2.30)
The interaction part appears unmodified also in the untilted ghost inflation case.
While the wave function part is characteristic of the wavefunction, and changes in
22
the two cases.
Our 3-point function can be approximately considered as a function of only two independent variables. The delta function, in fact, eliminates one of the three momenta,
imposing the vectorial sum of the three momenta to form a closed triangle. Because
of the symmetry of the De Sitter universe, the 3-point function is scale invariant, and
so we can choose [ki[ = 1. Using rotation invariance, we can choose k1 =
1,
and
impose k2 to lie in the 61, 62 plane. So, we have finally reduced the 3-point function
from being a function of 3 vectors, to be a function of 2 variables. From this, we
can choose to plot the 3-point function in terms of xi - ki, i = 1, 2. The result is
shown in fig.2-1. Note that we chose to plot the three point function with a measure
equal to x2x2. The reason for this is that this results in being the natural measure in
the case we wish to represent the ratio between the signal associated to the 3-point
function with respect to the signal associated to the 2-point function [5]. Because of
the triangular inequality, which implies
3 < 1 - x 2,
and in order to avoid to double
represent the same momenta configuration, we set to zero the three point function
outside the triangular region: 1 - x2 _
3 < x 2.
In order to stress the difference
with the case of standard ghost inflation, we plot in fig.2-2 the correspondent 3-point
function for the case of ghost inflation without tilt. Note that, even though the two
shapes are quite similar, the 3-point function of ghost inflation without tilt changes
signs as a function of the k's, while the 3-point function in the tilted case has constant
sign.
An important observation is that, in the limit as
3
-- 0 and x2 -
1, which
corresponds to the limit of very long and thin triangles, we have that the 3 point
function goes to zero as
roll inflation result
-
1 . This is expected, and in contrast with the usual slow
.
3
The reason for this is the same as the one which creates
the same kind of behavior in the ghost inflation without tilt [2]. The limit of x 3 -- 0
*
corresponds to the physical situation in which the mode k 3 exits from the horizon,
freezes out much before the other two, and acts as a sort of background.
In this
limit, let us imagine a spatial derivative acting on 7r3, which is the background in
the interaction Lagrangian. The 2-point function < 7r7r2 > depends on the position
23
1
0.8
Figure 2-1: Plot of the function F(l, X2, x3)x~x5 for the tilted ghost inflation 3-point
function. The function has been normalized to have value 1 for the equilateral configuration X2 = X3 = 1, and it has been set to zero outside of the region 1 - X2 :S X3 :S X2
0.5 F(x2,x3)
o
1
0.8
Figure 2-2: Plot of the similarly defined function F(l, X2, x3)x~x5 for the standard
ghost inflation 3-point function. The function has been normalized to have value 1
for the equilateral configuration X2 = X3 = 1, and it has been set to zero outside of
the region 1 - X2 :S X3 :S X2 [5]
on the background
wave, and, at linear order, will be proportional
variation of the 2-point function along the
1r3
24
to
8(Tr3.
The
wave is averaged to zero in calculating
the 3-point function < 7rk 7rk2 irk3 >, because the spatial average <
7r3 i7r3
> vanishes.
So, we are forced to go to the second order, and we therefore expect to receive a
factor of k, which accounts for the difference with the standard slow roll inflation
case. In the model of ghost inflation, the interaction is given by derivative terms,
which favors the correlation of modes freezing roughly at the same time, while the
correlation is suppressed for modes of very different wavelength. The same situation
occurs in standard slow roll inflation when we study non-gaussianities generated by
higher derivative terms [6].
The result is in fact very similar to the one found in [6]. In that case, in fact, the
interaction term could be represented as:
Lint
2(_,2 + e-2Ht(Oip)2)
(2.31)
where one of the time derivative fields is contracted with the classical solution. This
interaction gives rise to a 3-point function, which can be recast as:
< k(k2(k3 >a
((ki
)) ((k2 + k3)kt+ k2 +2k2k3) + cyclic)+ (2.32)
12
H f(ka)ka(k +
+ k3)
We can easily see that the first part has the same k dependence as our tilted ghost
inflation. That part is in fact due to the interaction with spatial derivative acting,
and it is equal to our interaction.
The integrand in the formula for the 3-point
function is also evaluated with the same wave functions, so, it gives necessarily the
same result as in our case. The other term is due instead to the term with three time
derivatives acting. This term is not present in our model because of the spontaneous
breaking of Lorentz symmetry, which makes that term more irrelevant that the one
with spatial derivatives, as it is explained in [2]. This similarity could have been
expected, because, adding a tilt to the ghost potential, we are converging towards
standard slow roll inflation. Besides, since we have a shift symmetry for the ghost
field, the interaction term which will generate the non gaussianities will be a higher
25
derivative term, as in [6].
We can give a more quantitative estimate of the similarity in the shape between
our three point function and the three point functions which appear in other models.
Following [5],we can define the cosine between two three point functions F1 (k1 , k 2, k 3 ),
F2(kl, k2, k3), as:
2
F2)1/
(F. F
cos(Fi, F2)
(2.33)
where the scalar product is defined as:
Fl(kl, k2, k3) . F2(k, k2, k3 )
d
/2
2
-x2
dx2
dx
F(1,x 2 ,x 3)F2 (1,x 2 ,X 3 ) (2.34)
where, as before, xi = k. The result is that the cosine between ghost inflation with
tilt and ghost inflation without tilt is approximately 0.96, while the cosine with the
distribution from slow roll inflation with higher derivatives is practically one. This
means that a distinction between ghost inflation with tilt and slow roll inflation with
higher derivative terms , just from the analysis of the shape of the 3-point function,
sounds very difficult. This is not the case for distinguishing from these two models
and ghost inflation without tilt.
Finally, we would like to make contact with the work in [7], on the Dirac-BornInfeld (DBI) inflation. The leading interaction term in DBI inflation is, in fact, of
the same kind as the one in (2.31), with the only difference being the fact that the
relative normalization between the term with time derivatives acting and the one
with space derivatives acting is weighted by a factor
2
=
(1
-
v2)- 1 , where v, is
the gravity-side proper velocity of the brane whose position is the Inflaton.
This
relative different normalization between the two terms is in reality only apparent,
since it is cancelled by the fact that the dispersion relation is w
k.
This implies
the the relative magnitude of the term with space derivatives acting, and the one
of time derivatives acting are the same, making the shape of the 3-point function in
DBI inflation exactly equal to the one in slow roll inflation with higher derivative
couplings, as found in [6].
26
2.3.3
Observational Constraints
We are finally able to find the observational constraints that the negative tilt in the
ghost inflation potential implies.
In order to match with COBE:
1
H
H4
152
)43M (4.8 10-5) 2
PC = 47r233/2
=
0.018l 83/ 6 3 /4
(2.35)
From this, we can get a condition for the visibility of the tilt. Remembering that
62
-
al/2
_3
(
), we get that, in order for 6 to be visible:
Mh
62
62 =
208
0.018 1/263/4
5/8
/~5/8
62 >
=4/5= 0.0016
2ibility
13
(2.36)
In the analysis of the data (see for example [8]), it is usually assumed that the
non-gaussianities come from a field redefinition:
=(
3
2
- 5fNL(C- <
>)
(2.37)
where Cgis gaussian. This pattern of non-gaussianity, which is local in real space, is
characteristic of models in which the non-linearities develop outside the horizon. This
happens for all models in which the fluctuations of an additional light field, different
from the inflaton, contribute to the curvature perturbations we observe. In this case
the non linearities come from the evolution of this field into density perturbations.
Both these sources of non-linearity give non-gaussianity of the form (2.37) because
they occur outside the horizon. In the data analysis, (2.37) is taken as an ansatz, and
limits are therefore imposed on the scalar variable
fNL.
The angular dependence of
the 3-point function in momentum space implied by (2.37) is given by:
<
(klk2k3
> = (27r)363(
ki)(2W)4(--fNL
PR)
Hi
k
(2.38)
In our case, the angular distribution is much more complicated than in the previous
expression, so, the comparison is not straightforward. In fact, the cosine between the
27
two distributions is -0.1. We can nevertheless compare the two distributions (2.27)
and (2.37) for an equilateral configuration, and define in this way an "effective"
fNL
for kl = k2 = k3. Using COBE normalization, we get:
fNL = -2
0.29
(2.39)
The present limit on non-gaussianity parameter from the WMAP collaboration [8]
gives:
-58 < fNL < 138 at 95% C.L.
(2.40)
and it implies:
62
> 0.005
(2.41)
which is larger than 52ivility (which nevertheless depends on the coupling constants
a,O0.
Since for
62 >> usiibility
we do see the effect of the tilt, we conclude that there is
a minimum constraint on the tilt: 62 > 0.005.
In reality, since the shape of our 3-point function is very different from the one
which is represented by
fNL,
it is possible that an analysis done specifically for this
shape of non-gaussianities may lead to an enlargement of the experimental boundaries.
As it is shown in [5], an enlargement of a factor 5-6 can be expected. This would lead
to a boundary on 62 of the order
62
> 0.001, which is still in the region of interest for
the tilt.
Most important, we can see that future improved measurements of Non Gaussianity in CMB will immediately constraint or verify an important fraction of the
parameter space of this model.
Finally, we remind that the tilt can be quite different from the scale invariant
result of standard ghost inflation:
]n,
1<
28
-
Ne
-1
(2.42)
2.4
Positive Tilt
In this section, we study the possibility that the tilt in the potential of the ghost
is positive, V' > 0. This is quite an unusual condition, if we think to the case of
the slow roll inflation. In this case, in fact, the value of H is actually increasing with
time. This possibility is allowed by the fact that, on the contrary with respect to what
occurs in the slow roll inflation, the motion of the field is not due to an usual potential
term, but is due to a spontaneous symmetry breaking of time diffeomorphism, which
gives a VEV to the velocity of the field. So, if the tilt in the potential is small enough,
we expect to be no big deviance from the ordinary motion of the ghost field, as we
already saw in section one.
In reality, there is an important difference with respect to the case of negative
tilt: a positive tilt introduces a wrong sign kinetic energy term for r. The dispersion
relation, in fact, becomes:
- p 2 k2
W2 =
(2.43)
The k2 term is instable. The situation is not so bad as it may appear, and the reason
is the fact that we will consider a De Sitter universe. In fact, deep in the ultraviolet
the term in k4 is going to dominate, giving a stable vacuum well inside the horizon.
As momenta are redshifted, the instable term will tend to dominate. However, there
is another scale entering the game, which is the freeze out scale w(k) - H. When this
occurs, the evolution of the system is freezed out, and so the presence of the instable
term is forgotten.
So, there are two possible situations, which resemble the ones we met for the
negative tilt. The first is that the term in k 2 begins to dominate after freezing out. In
this situation we would not see the effect of the tilt in the wave function. The second
case is when there is a phase between the ultraviolet and the freezing out in which
the term in k 2 dominates. In this case, there will be an instable phase, which will
make the wave function grow exponentially, until the freezing out time, when this
growing will be stopped. We shall explore the phase space allowed for this scenario,
29
which occurs for
3262
>
a 1/ 2 H
r=
a:/ H
(2.44)
and we restrict to it.
Before actually beginning the computation, it is worth to make an observation.
All the computation we are going to do could be in principle be obtained from the
case of positive tilt, just doing the transformation
-62 in all the results we
62 _-
obtained in the former section. Unfortunately, we can not do this. In fact, in the
former case, we imposed that the term in k2 dominates at freezing out, and then
solved the wave equation with the initial ultraviolet vacua defined by the term in k 2 ,
and not by the one in k4 , as, because of adiabaticity, the field remains in the vacua
well inside the horizon. On the other hand, in our present case, the term in k2 does
not define a stable vacua inside the horizon, so, the proper initial vacua is given by
the term in k4 which dominates well inside the horizon. This leads us to solve the
full differential equation:
u" + (_ 3 2k2+ a
k4H2 7 2
2
2
=
(2.45)
Since we are not able to find an analytical solution, we address the problem with
the semiclassical WKB approximation. The equation we have is a Schrodinger like
eigenvalue equation, and the effective potential is:
V = p2k 2 _ a k
4 H2 2
M2
+ -2
r2
(2.46)
Defining:
2 - /332 M 2
oh=to
Mr
(2.47)
we have the two semiclassical regions:
for
<< r7o,the potential can be approximated to:
4H2r2
¢ ~ -aM M22
30
(2.48)
(2.48)
while, for
> r70:
V= d,32 k2+
2
(2.49)
772
The semiclassical approximation tells us that the solution, in these regions, is given
by:
for r << 770:
U
(p( Al
(2.50)
ei fcr p(')d'
while, for 7r> r70:
U
(p
e-)crP(7)d7'
(2.51)
1 /2
where P(77)= (IV(77))
The semiclassical approximation fails for r7
ro. In that case, one can match the
two solution using a standard linear approximation for the potential, and gets A 2 =
Al
e -i
62
7r/4
[9]. It is easy to see that the semiclassical approximation is valid when
62.
Let us determine our initial wave function. In the far past, we know that the
solution is the one of standard ghost inflation [2]:
7r12
U=(
1
r(1) H k 2 a-
(-)1/2(
77)1/2H()
(2.52)
r2)
We can put this solution, for the remote past, in the semiclassical form, to get:
1
=
U
(2--k
/e e
(8+H
e 2M 77
(2.53)
(-77))1/2
So, using our relationship between Al and A 2 , we get, for r > ro, the following wave
function for the ghost field:
w = u/a
1
2/2
1/2ei(-
i _7
*
2He-")
2H e aH (Hrek" +
H
(2.54)
e-kr/)
(2.54)
Notice that this is exactly the same wave function we would get if we just rotated
5 -
i in the solutions we found in the negative tilt case. But the normalization
31
would be very different, in particular missing the exponential factor, which can be
large. It is precisely this exponential factor that reflects the phase of instability in
the evolution of the wave function.
From this observation, the results for the 2-point ant 3-point functions are immediately deduced from the case of negative tilt, paying attention to the factors coming
from the different normalization constants in the wave function.
So, we get:
2162M
1 e al
4P13/2H
H
(2.55)
4
(H
Notice the exponential dependence on a, 3, HIM, and 2.
The tilt gets modified, but the dominating term
n-1
= V2M 2
n=V'(HV
V"/
1
2
(-
+
4M
+
1-'is not modified:
27r3
62M +
a H2M
(2.56)
M
4
+
2
+
)
(1 - 2P
r 62
H
3
3
3H M2
MM
(2
- 4P"M
8
))
For the three point function, we get:
H8
<
k3
ki) 433a8M8
kl(k 2 (k3 >= (27r)363(Z
1
2 k3
(,62(,Z262
(kl2(.k3)
((k2 + k 3 )kt + kt + 2k3k2) + cyclic) e 6
(2.57)
H
which has the same k's dependence as in the former case of negative tilt. Estimating
the
fNL
as in the former case, we get:
fNL
-
0.29 6M
62 e
aH
(2.58)
(2.58)
Notice again the exponential dependence.
Combining the constraints from the 2-point and 3-point functions, it is easy to see
that a relevant fraction of the parameter space is already ruled out. Anyway, because
of the exponential dependenceon the parameters 62,- , and the coupling constants a,
and 3, which allows for big differences in the observable predictions, there are many
32
configurations that are still allowed.
2.5
Conclusions
We have presented a detailed analysis of the consequences of adding a small tilt to
the potential of ghost inflation.
In the case of negative tilt, we see that the model represent an hybrid between
ghost inflation and slow roll inflation. When the tilt is big enough to leave some signature, we see that there are some important observable differences with the original
case of ghost inflation. In particular, the tilt of the 2-point function of ( is no more
exactly scale invariant n, = 1, which was a strong prediction of ghost inflation. The
3-point function is different in shape, and is closer to the one due to higher deriva-
tive terms in slow roll inflation. Its total magnitude tends to decrease as the tilt
increases. It must be underlined that the size of these effects for a relevant fraction
of the parameter space is well within experimental reach.
In the case of a positive tilt to the potential, thanks to the freezing out mechanism, we are able to make sense of a theory with a wrong sign kinetic term for the
fluctuations around the condensate, which would lead to an apparent instability. Consequently, we are able to construct an interesting example of an inflationary model in
which H is actually increasing with time. Even though a part of the parameter space
is already excluded, the model is not completely ruled out, and experiments such as
WMAP and Plank will be able to further constraint the model.
33
Bibliography
[1] A. H. Guth, "The Inflationary Universe: A Possible Solution To The Horizon And
Flatness Problems," Phys. Rev. D 23 (1981) 347. A. D. Linde, "A New Inflationary
Universe Scenario: A Possible Solution Of The Horizon, Flatness, Homogeneity,
Isotropy And Primordial Monopole Problems," Phys. Lett. B 108 (1982) 389.
J. M. Bardeen, P. J. Steinhardt and M. S. Turner, "Spontaneous Creation Of
Almost Scale - Free Density Perturbations In An Inflationary Universe," Phys.
Rev. D 28 (1983) 679.
[2] N. Arkani-Hamed, P. Creminelli, S. Mukohyama and M. Zaldarriaga, "Ghost inflation," JCAP 0404 (2004) 001 [arXiv:hep-th/0312100].
[3] N. Arkani-Hamed,
H. C. Cheng, M. A. Luty and S. Mukohyama,
"Ghost conden-
sation and a consistent infrared modification of gravity," arXiv:hep-th/0312099.
[4] J. Maldacena, "Non-Gaussian features of primordial fluctuations in single field
inflationary models," JHEP 0305 (2003) 013 [arXiv:astro-ph/0210603]. See also
V.Acquaviva, N.Bartolo, S.Matarrese and A.Riotto,"Second-order cosmological
perturbations from inflation", Nucl.Phyis. B 667, 119 (2003) [astro-ph/0207295]
[5] D. Babich, P. Creminelli and M. Zaldarriaga, "The shape of non-Gaussianities,"
arXiv:astro-ph/0405356.
[6] P. Creminelli, On non-gaussianities in single-field inflation, JCAP 0310 (2003)
003 [arXiv:astro-ph/0306122].
34
[7] M. Alishahiha,
E. Silverstein and D. Tong, "DBI in the sky," arXiv:hep-
th/0404084.
[8] E. Komatsu et al., "First Year Wilkinson Microwave Anisotropy Probe (WMAP)
Observations:
Tests of Gaussianity,"
Astrophys. J. Suppl. 148 (2003) 119
[arXiv:astro-ph/0302223].
[9] See for example: Ladau, Lifshitz, Quantum Machanics, Vol.III, Butterworth
Heinemann
ed. 2000.
35
Chapter 3
How heavy can the Fermions in
Split Susy be? A study on
Gravitino and Extradimensional
LSP.
In recently introduced Split Susy theories, in which the scale of Susy breaking is
very high, the requirement that the relic abundance of the Lightest SuperPartner
(LSP) provides the Dark Matter of the Universe leads to the prediction of fermionic
superpartners around the weak scale. This is no longer obviously the case if the
LSP is a hidden sector field, such as a Gravitino or an other hidden sector fermion,
so, it is interesting to study this scenario. We consider the case in which the NextLightest SuperPartner (NLSP) freezes out with its thermal relic abundance, and then
it decays to the LSP. We use the constraints from BBN and CMB, together with the
requirement of attaining Gauge Coupling Unification and that the LSP abundance
provides the Dark Matter of the Universe, to infer the allowed superpartner spectrum.
As very good news for a possible detaction of Split Susy at LHC, we find that if the
Gravitino is the LSP, than the only allowed NLSP has to be very purely photino like.
In this case, a photino from 700 GeV to 5 TeV is allowed, which is difficult to test
36
at LHC. We also study the case where the LSP is given by a light fermion in the
hidden sector which is naturally present in Susy breaking in Extra Dimensions. We
find that, in this case, a generic NLSP is allowed to be in the range 1-20 TeV, while
a Bino NLSP can be as light as tens of GeV.
3.1
Introduction
Two are the main reasons which lead to the introduction of Low Energy Supersymmetry for the physics beyond the Standard Model: a solution of the hierarchy problem,
and gauge coupling unification.
The problem of the cosmological constant is usually neglected in the general treat-
ment of beyond the Standard Model physics, justifying this with the assumption that
its solution must come from a quantum theory of gravity. However, recently [1], in
the light of the landscape picture developed by a new understanding of string theory,
it has been noted that, if the cosmological constant problem is solved just by a choice
of a particular vacua with the right amount of cosmological constant, the statistical
weight of such a fine tuning may dominate the fine tuning necessary to keep the Higgs
light. Therefore, it is in this sense reasonable to expect that the vacuum which solves
the cosmological constant problem solves also the hierarchy problem.
As a consequence of this, the necessity of having Susy at low energy disappears,
and Susy can be broken at much higher scales (106 - 109 GeV).
However, there is another important prediction of Low Energy Susy which we
do not want to give up, and this is gauge coupling unification. Nevertheless, gauge
coupling unification with the same precision as with the usual Minimal Supersymmetric Standard Model (MSSM) can be achieved also in the case in which Susy is
broken at high scales. An example of this is the theories called Split Susy [1, 2] where
there is an hierarchy between the scalar supersymmetric partners of Standard Model
(SM) particles (squarks, sleptons, and so on) and the fermionic superpartners of SM
particles (Gaugino, Higgsino), according to which, the scalars can be very heavy at
37
an intermediate scale of the order of 109 GeV, while the fermions can be around the
weak scale. The existence for this hierarchy can be justified by requiring that the
chiral symmetry protects the mass of the fermions partners.
While the chiral symmetry justifies the existence of light fermions, it can not fix
the mass of the fermionic partners precisely at the weak scale. As a consequence,
this theory tends to make improbable the possibility of finding Susy at LHC, because
in principle there could be no particles at precisely 1 TeV. In this chapter, for Split
Susy, we do a study at one-loop level of the range of masses allowed by gauge coupling
unification, finding that these can vary in a range that approximately goes up to
20 TeV. A possible way out from this depressing scenario comes from realizing that
cosmological observations indicate the existence of Dark Matter (DM) in the universe.
The standard paradigm is that the Dark Matter should be constituted by stable
weakly interacting particles which are thermal relics from the initial times of the
universe. The Lightest Supersymmetric Partner (LSP) in the case of conserved Rparity is stable, and, if it is weakly interacting, such as the Neutralino, it provides a
perfect candidate for the DM. In particular, an actual calculation shows that in order
for the LSP to provide all the DM of the universe, its mass should be very close to
the TeV scale. This is the very good news for LHC we were looking for. Just to stress
this result, it is the requirement the the DM is given by weakly interacting LSP that
forces the fermions in Split Susy to be close to the weak scale, and accessible at LHC.
In three recent papers [2, 3, 4], the predictions for DM in Split Susy were investigated, and revealed some regions in which the Neutralino can be as light as
200
GeV (Bino-Higgsino), and some others instead where it is around a 1 TeV (Pure
Higgsino) or even 2 TeV (Pure Wino). As we had anticipated, all these scales are
very close to one TeV, even though only the Bino-Higgsino region is very good for
detection at LHC.
Since the Dark Matter Observation is really the constraint that tells us if this kind
of theories will be observable or not at LHC, it is worth to explore all the possibilities
for DM in Split Susy. In particular, a possible and well motivated case which had
been not considered in the literature, is the case in which the LSP is a very weakly
38
interacting fermion in a hidden sector.
In this chapter, we will explore this possibility in the case in which the LSP is
either the Gravitino, or a light weakly interacting fermion in the hidden sector which
naturally appears in Extra Dimensional Susy breaking models of Split Susy [1, 5].
We will find that, if the Gravitino is the LSP, than all possible candidates for the
NLSP are excluded by the combination of imposing gauge coupling unification and
the constraint on hadronic decays coming from BBN. Just the requirement of having
the Gravitino to provide all the Dark Matter of the univese and to still have gauge
coupling unification would have allowed weakly interacting fermionic superpartneres
as heavy as 5 TeV, with very bad consequences on the detactibility of Split Susy at
LHC. This means that these constraints play a very big role. The only exception to
this result occurs if the NLSP is very photino like, avoiding in this way the stringent
constraints on hadronic decays coming from BBN. However, as we will see, already a
small barionic decay branching ratio of 10- 3 is enough to rule out also this possibility.
For the Extradimensional LSP, we will instead find a wide range of possibilities,
with NLSP allowed to span from 30 GeV to 20 TeV.
The chapter is organized as follows. In section 4.2, we study the constraints on the
spectrum coming from the requirement of obtaining gauge coupling unification. In
section 4.3, we briefly review the relic abundance of Dark Matter in the case the LSP
is an hidden sector particle. In section 4.4, we discuss the cosmological constraints
coming from BBN and CMB. In section 4.5, we show the results for Gravitino LSP.
In section 4.6, we do the same for a dark sector LSP arising in extra dimensional
implementation of Split Susy. In section 4.7, we draw our conclusions.
3.2
Gauge Coupling Unification
Gauge coupling unification is a necessary requirement in Split Susy theories. Here we
investigate at one loop level how heavy can be the fermionic supersymmetric partner
for which gauge coupling unification is allowed. We will consider the Bino, Wino,
and Higgsino as degenerate at a scale M 2, while we will put the Gluinos at a different
39
scale M 3 .
Before actually beginning the computation, it is interesting to make an observation
about the lower bound on the mass of the fermionic superpartners. Since the Bino is
gauge singlet, it has no effect on one-loop gauge coupling unification. In Split Susy,
with the scalar superpartners very heavy, the Bino is very weakly interacting, its only
relevant vertex being the one with the light Higgs and the Higgsino. This means that,
while for the other supersymmetric partners LEP gives a lower bound of
-
50-100
GeV [11], for the Bino in Split Susy there is basically no lower limit.
Going back to the computation of gauge coupling unification, we perform the
study at -loop level. The renormalization group equations for the gauge couplings
are given by:
A
dgi
1
dA
(4ir) 2
bi(A)g
3
(3.1)
where bi(A) depends of the scale, keeping truck of the different particle content of
the theory according to the different scales, and i = 1, 2, 3 represent respectively
/-5/3g',g, g,. We introduce two different scales for the Neutralinos, M 2, and for the
Gluinos M3 , and for us M3 > M2.
In the effective theory below M2 , we have the SM, which implies:
41
19
bM =(10' -- 7 -7)
(3.2)
Between M 2 and M3:
bsplitl
= (
9
7
2'
6'
7)
(3.3)
Between M3 and mh,which is the scale of the scalars:
2 = (
bsplit
9
7
2'
6
,
-5)
(3.4)
and finally, above ri we have the SSM:
bssm(33
bssm
33( 1, -3)
5
40
(3.5)
The way we proceed is as follows: we compute the unification scale MGUT and
aGUT as deduced by the unification of the SU(2) and U(1) couplings. Starting from
this, we deduce the value of as at the weak scale Mz, and we impose it to be within
the 2 experimental result aS(Mz) = 0.119 ± 0.003. We use the experimental data:
sin2 (Ow(MIz))= 0.23150± 0.00016and a- 1(Mz) = 128.936± 0.0049[12].
A further constraint comes from Proton decay p -+ lrOe+,which has lifetime:
8fM
2M4
T((1 + GrmP
DUT F)AN) 2
T(p -- roe+) =
(
MGUT
1/35\
10 16 GeV
aGUT
2
0.15GeV
\
(3.6)
\ 1.3 x 1035 yr
CaN
where we have taken the chiral Lagrangian factor (1 + D + F) and the operator
renormalization A to be (1 + D + F)A - 20. For the Hadronic matrix element aN,
we take the lattice result [13] acN = 0.015GeV3 . From the Super-Kamiokande limit
[14], r(p -
r°e+ ) > 5.3 x 1033 yr, we get:
MGUT> 0.01
V3)
1/2
12GUT
GUT/35 4 x 1015 GeV
(3.7)
An important point regards the mass thresholds of the theory. In fact, the spectrum of the theory will depend strongly on the initial condition for the masses at
the supersymmetric scale im. As we will see, in particular, the Gluino mass M3 has
a very important role for determining the allowed mass range for the Next-Lightest
Supersymmetric Particle (NLSP), which is what we are trying to determine. In the
light of this, we will consider M2 as a free parameter, with the only constraint of being
smaller than mn. M 3 will be then a function of M2 and fm, and its actual value will
depend on the kind of initial conditions we require. In order to cover the larger fraction of parameter space as possible, we will consider two distinct and well motivated
initial conditions. First, we will require gaugino mass unification at m. This initial
condition is the best motivated in the approach of Split Susy, where unification plays
a fundamental role. Secondarily, we will require anomaly mediated gauigino mass initial conditions at the scale m. This second kind of initial conditions will give results
41
quite different from those of Gaugino mass unification, and, even if in this case the
Gravitino can not be the NLSP, the field Ix, which will be a canditate LSP from
extradimensions that we will introduce in the next sections, could be still the LSP.
3.2.1
Gaugino Mass Unification
Here we study the case in which we apply gaugino mass unification at the scale 7i.
In [2], a 2-loop study of the renormalization group equations for the Gaugino mass
starting from this initial condition was done, and it was found that, according to mn
and M2, the ratio between M 3 and M2 can vary in a range
-
3 - 8. We shall use
their result for M3 , as the value of M3 will have influence on the results, tending to
increase the upper limit on the fermions' mass.
At one loop level, we can obtain analytical results. After integration of eq.(3.1),
we get the following expressions:
MGUT =
(e
(M(b
-bn
9
) M((bMPvit1
-blm)_(bsplitl-bS))
2
1--
2
In M-
(1b~pitl+b2m)
(3.8)
(38)
1pit
M(bit2
split23
bMbi) valuesm
2 (bSsm +bplit2)
-"
turns
loops
out-(b
that
are 8w
important
to determbine
the predicted
2
gTbItltwo
g 2 effect
(Mzu)
(bssm+bsPt
\bssm
(bsPit 2 +bsplitl)
of aoe(Mz). Since our main purpose is to have a rough idea of the maximum scale
for the fermionic masses allowed by Gauge Coupling Unification, we proceed in the
42
following way. In [2], 2-loop gauge coupling unification was studied for M 2 = 300 GeV
and 1 TeV. Since the main effect of the 2-loop contribution is to raise the predicted
value of a,(Mz),
we translate our predicted value of as(Mz) to match the result in
[2] for the correspondent values of M 2. Having set in this way the predicted scale for
a,(Mz),
we check what is the upper limit on fermion masses in order to reach gauge
coupling unification. The amount of translation we have to do is: 0.008.
In fig.3-1, we plot the prediction for a,(Mz) for M2 = 300 GeV, 1 TeV, and 5
TeV. We see that for 5 TeV, unification becomes impossible. And so, 5 TeV is the
upper limit on fermionic superpartner allowed from gauge coupling unification. Note
that the role of the small difference between M3 and M2 is to raise this limit.
as (Mz)
U.1
0 .125
IVe
0.115
Tev
6
8
10
12
14
Log(ffi/GeV)
Figure 3-1: In the case of gaugino mass unification at scale ms,we plot the unification
prediction for a,(Mz). The results for M 2 = 300 GeV, 1 TeV and 5 TeV are shown.
The horizontal lines represent the experimental bounds
In fig.3-2, and fig.3-3,we plot the predictions for CYGUT(MGUT)and for MGUT,
for the same range of masses. We see that unification is reached in the perturbative regime, with unification scale large enough to avoid proton decay limits. Note,
however, that for M2 = 5 TeV, the limit is close to a possible detection.
Finally, note that with this Gaugino mass initial conditions, the Wino can not be
the NLSP if the Gravitino is the LSP, as shown in [2].
43
a (UT)
0
0
0
0
Log (m/GeV)
0
0
Figure 3-2: In the case of Gaugino mass unification at scale h, we plot the prediction
for ca,(MGUT). The results for M2 = 300 GeV, 1 TeV and 5 TeV are shown.
"GUT
-
-
16
Z-1U
1.75.1016
1.5
1016
1.25 .1016
1.1016
7.5 -56
- 1 1-44,
8
10
12
14
16
Log Cn/GeV)
MI imit
Figure 3-3: In the case of Gaugino mass unification at scale mi, we plot the prediction
for MGUT. The results for M2 = 300 GeV, 1 TeV and 5 TeV are shown, together with
the lower bound on MGUTfrom Proton decay.
As we will see later, a particular interesting case for the LSP in the hidden sector
is given by a Bino NLSP. For this case, we need to do a more accurate computation,
splitting the mass of the Gauginos, from that of the Higgsinos,and taking the Wino
mass roughly two times larger than the Bino mass, as inferred from [2] for gaugino
mass unification initial conditions. In fig.3-4, we show what is the allowed region for
44
the mass of the Bino and the ratio of the Hissino mass and Bino mass, such that
gauge coupling unification is attained with a mass for the scalars, m, in the range
105 GeV-10 8 GeV. Raising the Higgsino mass with respect to the Bino mass has the
effect of lowering the maximum mass for the fermionic superpartners. This is due to
the fact that, raising the Higgsino mass, the unification value for the U(1) and SU(2)
couplings is reduced, so that the prediction for a,(Mz) is lowered.
MB
MS
Log(M/GeV)
Figure 3-4: Shaded is the allowed region for the Bino mass and the ratio of the
Higgsino mass and the Bino mass, in order to obtain Gauge Coupling Unification
with a value of the scalar mass rh in the range 105 GeV-101 8 GeV. We take M2 ~- 2M1
as inferred from gaugino mass unification at the GUT scale [2]
3.2.2
Gaugino Mass Condition from Anomaly Mediation
Of the possible initial conditions for the Gaugino mass which can have some influence
on the upper bound on fermions mass, there is one which is particularly natural, and
which is coming from Anomaly Mediated Susy breaking, and according to which the
initial conditions for the gaugino masses are:
Mi = gi 3/23/2 ' i-'
M, g~ia
lxm
45
(3.11)
(3.11)
where 3i is the beta-function for the gauge coupling, and ci is an order one number.
These initial conditions are not relevant for the Gravitino LSP, as in this case the
Neutralinos are lighter than the Gravitinos; but they can be relevant in the case the
LSP is given by a fermion in the hidden sector, as we will study later.
Further,
the study of this case is interesting on its own, as it gives an upper bound on the
fermionic superpartners which is higher with respect to the one coming from gaugino
mass unification initial conditions.
The study parallels very much what done in the former section, with the only
difference being the fact that in this case, as computed in [2], the mass hierarchy
between the Gluinos and the Gauginos is higher ( a factor -
10 - 20 instead of
3 - 8). This has the effect of raising the allowed mass for the fermions. We do the
same amount of translation as before for the predicted a,(Mz).
The result is shown
in fig.3-5, and gives, as upper limit, M2 = 18 TeV.
aS(M )
tx
19a
V . 1
300 GeV
0.125
1TeV
0.12
1 TeV
0.115
.
.
.
6
I8T
eV....................
.......
8
10
12
14
Log(ffGeV)
Figure 3-5: In the case of Gaugino mass condition from anomaly mediation at scale
fi, we plot the unification prediction for a(Mz). The results for M2 = 300 GeV, 1
TeV and 18 TeV are shown. The horizontal lines represent the experimental bounds
In fig.3-7, and fig.3-6, we plot the predictions for
aGuT
and MGUT for the same
range of masses, and we see that unification is reached in the perturbative regime,
46
and that the unification scale is large enough to avoid proton decay limits, but it is
getting very close to the experimental bound for large values of the mass i.
( MGUT )
,.~ ,. A9-
b
8
1U
Log (m/GeV)
J
.
Figure 3-6: In the case of Gaugino mass from Anomaly Mediation, we plot the prediction for CCUT.The results for M 2 = 300 GeV, 1 TeV and 18 TeV are shown.
MGUT
, 16
Z I'IU
-
1.75.1016
18 TeV
1.5
1016
1.25
1016
1 TeV
1 1016
7.5.1015
=
I :),I
Z.
y
45
' 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11.1.1.1
6
12
14
......
16
Log (~/GeV)
MLimit
Figure 3-7: In the case of Gaugino mass from Anomaly Mediation, we plot the prediction for MGUT. The results for M2 = 300 GeV, 1 TeV and 18 TeV are show, together
with the lower bound on MGUT from Proton decay.
As we can see, in the case of Gaugino Mass from Anomaly Mediation, the upper
limit on fermion mass is raised to 18 TeV. This last one can be interpreted as a sort
47
of maximum allowed mass for fermionic superpartners.
It is important to note that, as pointed out in [2], in this case the Bino can not
be the NLSP.
3.3
Hidden sector LSP and Dark Matter Abundance
An hidden sector LSP which is very weakly interacting can well be the DM from the
astrophysical and cosmological point of view. Its present abundance can be given by
two different sources: it can be a thermal relic, if in the past the temperature was so
high that hidden sector particles were in equilibrium with the thermal bath, or it can
be present in the universe just as the result of the decay of the other supersymmetric
particles.
We concentrate in the case in which the thermal relic abundance is negligible,
which is generically the case for not too large reheating temperatures, and the abundance is given by the decaying of the other supersymmetric particles into the LSP. A
discussion on the consequences of a thermal relic abundance of Gravitinos is discussed
in [6].
In our case, the relic abundance of the heavier particles is what determines the
final abundance of the LSP, and so it is the fundamental quantity to analyze. In the
very early universe, the typical time scale of the cosmic evolution H - 1 is much larger
than the time scale of interaction of a weakly interacting particle, and so a weakly
interacting particle is in thermal equilibrium. Therefore, its abundance is given by
the one of a thermal distribution.
As the temperature of the universe drops down,
the interaction rate is not able anymore to keep the particle in thermal equilibrium,
and so the particle decouples from the thermal bath, and its density begins to dilute,
ignoring the rest of the thermal bath. We say in this case that the particle species
freezes out.
In the case of weakly interacting particles around the TeV scale, the freeze out
48
temperature is around decades of GeV, and so they are non relativistic at the moment
of freezing out. In this case, the relic abundance of these particles is given by the
following
formula
[7, 8, 9]:
(1019GeV)
J (30) 0 0i5
( 10-9GeV-2)
(15
< Uv >
(3.12)
where < av > is the thermally averaged cross section at the time of freeze out,
Xf
=
mNLsP
Tf
where Tf is the freeze out temperature, g is the effective number of
degrees of freedom at freeze out, and h is the Hubble constant measured in units of
100Km/(sec Mpc). It is immediate to see that, for weakly interacting particle at 1
TeV, the resulting Q is of order unity, and this has led to the claim that the Dark
Matter bounds some supersymmetric partners to be at TeV scale. In this chapter,
we shall check this claim for an LSP in the hidden sector.
Once the weakly interacting particles are freezed out, they will rapidly decay to
the NLSP, which, being lighter, will be in general still in thermal equilibrium. So,
it will be the NLSP the only one to have a relevant relic abundance, determined by
the freeze out mechanism, and so it will be the NLSP that, through its decay, will
generate the present abundance of the LSP.
In Split Susy, the NLSP can either be the lightest Neutralino, or the lightest
Chargino. The Neutralino is a mixed state of the interaction eigenstates Bino, Wino,
and neutral Higgsinos, and is the lightest eigenstate of the following matrix[3]:
_0kM1v kI v
0
\/-
kv
M2
k2 v
0
\/~
klv
V8-v
k1 v
-/i
8(3.13)
0
which differs from the usual Neutralino matrix in low energy Susy for the Yukawa
coupling, which have their Susy value at the Susy breaking scale m1,but then run
differently from that scale to the weak scale.
The Chargino is a mixed eigenstate of charged Higgsino and charged Wino, and
49
is the lightest eigenstate of the following matrix:
(M2 kLv
2
(3.14)
j/
The actual and precise computation of the thermally averaged cross section of
the NLSP at freeze out, which determines the QNLSP, is very complicated, because
there are many channels to take care of, which depend on the abundance of the
particles involved, and on the mixing of states, creating a very complicated system
of differential equations. A software called DarkSusy has been developed to reliably
compute the relic abundance [10], and, in a couple of recent papers [2, 3], it has been
used to compute the relic abundance of the Neutralino NLSP in Split Susy. In this
kind of theories, in particular, this computation is a bit simplified, since the absence
of the scalar superpartners makes many channels inefficient. Nevertheless, in most
cases, the computation is still too complicated to be done analytically.
In the study in this chapter, we consider both the possibility that the NLSP is a
Neutralino and a Chargino. In the case of Neutralino NLSP, we modify the Dark Susy
code [10] and adapt it to the case of Split Susy. We consider the cases of pure Bino,
pure Wino, pure Higgsino, and Photino NLSP. In the case of Chargino NLSP, we
consider the case of charged Higgsino and Charged Wino as NLSP, and we estimate
their abundance with the most important diagram. We will see, in fact, that in this
case a more precise determination of the relic abundance is not necessary.
Once the NLSP has freezed out, it will dilute for a long time, until, at the typical
time scale of 1 sec, it will decay gravitationally to the LSP, which will be stable, and
will constitute today's Dark Matter. It's present abundance is connected to the NLSP
"would be" present abundance by the simple relation:
QLSP
-
L'LSP
m s NLSP
mNLSP
(3.15)
Already from this formula, we may get some important information on the masses
of the particles, just comparing with the case in which the Neutralino or the Chargino
50
is the LSP. In fact, since
mLsP
<
1,
QNLSP
has to be greater than what it would
have to have if the NLSP was the LSP, in order for the LSP to provide all the DM.
The abundance of the NLSP is inversely proportional to < av >, and this means
that we need to have a typical cross section smaller than the one we would obtain in
the case of a weakly interacting particle at TeV scale. This result can be achieved in
two ways: either raising the mass of the particles, since or
2, or by choosing some
particle which for some reason is very low interacting.
The direction in which the particle become very massive is not very attractive from
the LHC detection point of view, but still, in Split Susy, is in principle an acceptable
scenario.
The other direction instead immediately lets a new possible candidate to emerge,
which could be very attractive from the LHC detection point of view. In fact, in
Split Susy, a pure Bino NLSP is almost not interacting, the only annihilation channel
being the one into Higgs bosons in which an Higgsino is exchanged. In this case the
relic abundance has
QNLSP
> 1, and this was the reason why a pure Bino could not
be the DM in Split Susy [2, 3]. In the case of a gravitationally interacting LSP, as we
are considering, this over abundance would go exactly into the right direction, and it
could open a quite interesting region for detection at LHC.
3.4
Cosmological Constraints
Since we wish the LSP to be the Dark Matter of the universe, so, we impose its
abundance to cope with WMAP data [15, 16].
In general, for low reheating temperature, only the weakly interacting particles
are thermally produced, and only the NLSP will remain as a thermal relic in a relevant amount. Later on, it will decay to the LSP. This decay will give the strongest
cosmological constraints.
In fact, concentrating on the Gravitino, which interacts only gravitationally, we
51
can naively estimate its lifetime as:
r
3
MNLS
(3.16)
where the mNLSP term comes from dimensional analysis. In reality, we can easily
do better. In fact, as we have Goldstone bosons associated to spontaneous symmetry breaking, the breaking of supersymmetry leads to the presence of a Goldstino,
a massless spinor. Then, as usually occurs in gauge theories, the Goldstino is eaten
by the massless Gravitino, which becomes a massive Gravitino with the right number of polarization.
Therefore, the coupling of the longitudinal components of the
Gravitino to the LSP will be determined by the usual pattern of spontaneous symmetry braking, and in particular will be controlled by the scale of symmetry breaking.
This means that the coupling constant may be amplified. In fact, if we concentrate
on the Gauginos for simplicity, we can reconstruct their coupling to the Goldstino
simply by looking at the symmetry braking term in the lagrangian in unitary gauge,
then reintroducing the Goldstino performing a Susy transformation, and promoting
the transformation parameter to a new field, the Goldstino. The actual coupling is
then obtained after canonical normalization of the Goldstino kinetic term, which is
obtained performing a Susy transformation of the mass term of the Gravitino, and
remembering that in the case of Sugra, the Susy transformation of the Gravitino
contains a piece proportional to the vacuum energy. In formulas, the Gaugino Susy
transformation
is given by:
X = o4"Fu,
where
(3.17)
is the Goldstino. This implies that the mass term of the Gaugino sources the
following coupling between the Gaugino and the Goldstino:
6(mAA) D m\XAu"LF,,v
52
(3.18)
The Goldstino kinetic term cames from the Gravitino tranformation, which is:
6Pl = mppl
+ ifae
(3.19)
so, the Gravitino mass term produces the Goldstino kinetic term:
5(mgr,,ut, vbv>)D mgrfmpl
Z = f2maucr
(3.20)
where in the last expression we used that mgr = f/rmpl. So, after canonical normalization, we get the following interaction term:
LI = f2
.A1F.C
f2
where c =
(3.21)
f is the canonically normalized Goldstino. After all this, we get an
enhanced decay width like this:
M2
mgr)
rMNLSP
(3.22)
Note that this is independent on the particle species, as it must be by the equivalence
principle [17].
Plugging in some number, we immediately see that, for particles around the TeV
scale, without introducing a big hierarchy with the Gravitino, the time of decay is
approximately
1-sec, and this is right the time of Big Bang Nucleosynthesis (BBN).
This is the origin of the main cosmological bound. In fact, the typical decay of the
LSP will be into the Gravitino and into its SM partner. The SM particle will be very
energetic, especially with respect to a thermal bath which is of the order of 1 MeV,
and so it will create showers of particles, which will destroy some of the existent
nuclei, and enhance the formation of others, with the final result of alterating the
final abundance of the light elements [17].
There is also another quantity which comes into play, and it is what we can call
the "destructive power". In fact, the alteration of the light nuclei abundance will
53
be proportional to the product of the abundance of the decaying particle and to the
energy release per decay. This information is synthesized in an upper limit on the
variable ~ defined as:
=BEY
B
(3.23)
where B is the branching ratio for hadronic or electromagnetic decays (it turns out
that hadronic decays impose constrains a couple of order of magnitude more stringent
that electromagnetic decays),
is the energy release per decay, and finally Y =
n
where n is the number of photons per comoving volume, and nx the number of
decaying particles per comoving volume. Again, it is easy to see what will be the
lower limit on the upper limit on 6. For the moment, we will neglect the dependence
on the branching ratio B, because, clearly, one of the two branching ratios must be
of order one. Then, we understand that the most dangerous particles for BBN will
be those particles that decay when BBN has already produced most of the nulcei
we have to see today. A particle which decays earlier than this time, will in general
have its decay products diluted and thermalized with an efficiency that depends on
the kind of decay product of the particle: either baryonic or electromagnetic, and it
turns out that the dilution for electromagnetic decays is much more efficient. So, it
is clear that the upper limit on the "desctructive power"
will be lower for particles
which decay after BBN. For these late decaying particles, we can estimate what the
upper limit on
should be with the following argument.
will become dangerous if
the energy release is bigger than I MeV, in order for the decay product to be able to
destroy nuclei, and also if Y is greater than ni which represent the number of baryons
per comoving volume opportunely normalized. Plugging in the numbers, with, again,
B
-
1, we get
(dangerous
> 10-
14
GeV
(3.24)
This values of dangerous is more or less where the limit seems to apply in numerical
simulations for late decaying particles, and it is in fact independent on the particular
kind of decay, as in this case there are not dilution issues [18, 193. For early decaying
particles the limits do depend on the kind of decay, and they get more and more
54
relaxed as the decay time becomes shorter and shorter, until there is practically no
limit on particles which decay earlier than
10-2 sec. Notice however that, from
the estimates above on the decay time, the particles which we will be interested in
will tend to decay right in the region where these limits apply. The limit in eq.(3.24)
translates into another useful parameter:
Qdangerous
10 - 7
(3.25)
for the contribution of the NLSP around the time of nucleosynthesis. An easy computation shows that, imposing tQDM 1 today, we get that the contribution of NLSP
goes as:
QNLSP
1 0 -7MNLSP
(3.26)
(MeV
This estimates are obviously very rough, but they are useful to give an idea of the
physics which is going by, and they are, at the end of the day, quite accurate. They
nevertheless tell us that we are really in the region in which these limits are effective,
with two possible consequences: on one hand, a big part of the parameter region
might be excluded, but also, on the other hand, this tells us that a possible indirect
detection through deviations from the standard picture nucleosynthesis might reveal
new physics.
In two recent papers [18, 19], numerical simulation were implemented to determine
the constraints on , both for the hadronic and the electromagnetic decays, and we
shall use their data. (See also [20, 21] where a similar discussion is developed.)
Cosmological constraints come also from another observable.
A late decaying
particle can in fact alter the thermal distribution of the photons which then will form
the CMB, introducing a chemical potential in the CMB thermal distribution bigger
than the one which is usually expected due to the usual cosmic evolution, or even
bigger than the current experimental upper bound [22, 23]. Analytical formulas for
the produced effect are given in [24, 25]. Nevertheless, it is useful to notice that the
induced chemical potential
is mostly and hugely dependent on the time of decay of
55
the particles. In particular, we see that:
e
where
NLSP
e
(3.27)
TNLSP
is the lifetime of the NLSP, and rdc,
106 S is the time at which the
double Compton scattering of the photons is no more efficient. The
quantity is
dangerous,
dangerous
10 - 9 GeV. So, we conclude that basically, for
there are no limits, while for
rNLSP
TNLSP
for this
<
Td
,
> rd, the limit from nucleosynthesis is stronger.
We easily see that this constraint never comes into play in our work.
From formula (3.22), we can already extract an idea of what will be the final result
of the analysis. In fact, we can avoid the limits from nucleosynthesis by decaying early.
This means that, according to (3.22), we need to let the ratio
consequently
QNLSP
mNLSP
to grow, and
has to grow as well. This leads to two directions: either a very
massive LSP or a a very weakly interacting LSP. The first direction goes in agreement
with one of the directions we had found in order to match the constraint from QDM,
and tells us that, in general, a massive NLSP will be acceptable from the cosmological
point of view. However, it will have chanches to encounter the constraints coming
from gauge couplings unification. The other direction is to have an NLSP whose main
annihilation channel is controlled by another particle, which can be made heavy. As
an example, this is the case for the Bino, whose channel is controlled by the Higgsino:
so, we might have a light Bino, if the Higgsino will be heavier.
3.5
Gravitino LSP
In this section, we concentrate in detail on the possibility that the LSP in Split Susy
is the Gravitino, and that it constitute the Dark Matter of the universe. We shall
consider the mass of the Gravitino as a free parameter, and we shall try to extract
information on the mass and the nature of the NLSP. However, an actual lower limit
on the Gravitino mass can be expected in the case Susy is broken directly, as in that
case the mass of the Gravitino should be:
mgr
56
- M
, where h is the Susy breaking
MEl'
scale. Since, roughly, in Split Susy in is as light as
100 TeV, we get the lower limit
mgr > 10-8 GeV, which, as we will see, is lower than the region we will concentrate
on.
As we learnt in the former two sections, there are two fundamental quantities to
be computed: the lifetime of the NLSP, and QNLSP.
As we said before, we shall consider both the Neutralino and the Chargino as
LSP. The decaying amplitude of a Neutralino into Gravitino plus a Standard Model
particle was computed in [26, 27, 28].
For decay into Photons:
where X = Nnl(-iB) + N1 2 (-iW) + Nl 3Hd + N14 H is the NLSP. As we see,
eq.(3.22) reproduces the right behavior in the limit mNLSp/mLSp
> 1 . This decay
will contribute only to ElectroMagnetic (EM) energy.
The leading contribution from hadronic decays comes from the decay into Z, gr
and h, gr. These decays will contribute to EM or Hadronic energy according to the
branching ratios of the SM particles. The decay width to Z boson is given by:
r(X
Zgr)
-,
-
2
Zogr)) I~i
sin() N2CO(6)1
G(mwhere
=Z, gr)=
(X X
--)mgrn(-iz)
+
5x-F(mx7mgrrz)
(3.29)
3
4
2
/
M(-iT
+ 1id + N1 - 2 is
see,
2\, the NLSP. As
,) we
(331"
48r
M
m gr
2r) 21 + 3Msg) rnZ G(mx, mgr mz))
-
-
where
F(m mgrm )= ((1-(m 9 r+MZ)t
x
x
57
(3.30)
(1 ( mgr(
-mz-
x
x
x
The decay width to h boson is given by:
h, gr)
(X
+N s
_m
~ ~ ~- N1
~ sin(0)qM
r(~~X
3
-
m
x )m
where h =-H
x
(3.32)
F(m, gr
, mT) (3.32)
mr2
x Fcos(/3)mx
14
m2
21
487
mgr
)
sin(i3) + H°Cos(3) is the fine tuned light Higgs, and
2
H(mxMgrm h) =3+4
3
r+ 2 mgr+
mx
m
m
2
r+
gr+
m
3
mx
mx
2
2\
gr+3 g
mx
mx
(3.33)
EM
Bx
Had
F(
-
E
2
mx
mr-
1(3.35)
2m
h, gr)Bhad+ F(X --+q, q, gr)
Z, g)Bad + (X ,
+ r(x
Fr(x -> gyr)
x
22
2
mX
+ rF(
h, gr)
2
m gr
-m
2
Z,h
Z, gr)
(3.36)
(3.37)
and Bx is the branching ratio in the EM and Hadronic channel. We use Bhad
0.9, Bzaa
0.7. Since it will not play an important role, we just estimate the channel
(X - q, , gr), and do not perform a complete computation. This channel provides
the hadronic decays when mNLSP - mgr is less than the mz or ma. The leading
diagram in this case is given by the tree level diagram in which there is a virtual Z
boson or a virtual Higgs that decays into quarks.
3.5.1
Neutral Higgsino, Neutral Wino, and Chargino NLSP
An Higgsino NLSP will be naturally muchsyinteracting in Split Susy, quite independently of the other partners mass. In fact, there are gauge interaction and Yukawa
58
coupling to the other particles. While the coupling to the Z vanishes for
> mz, in
that case neutral Higgsino and Charged Higgsino are almost degenerate, and so the
interaction with the Higgs become relevant. This means that the annihilation rate
will never be very weak, and so QNLSP will be large only for large u. An analytical
computation is too complicated for our necessities, even with the simplifications of
Split Susy, so, we modify the DarkSusy code [10] to adapt it to the Split Susy case,
and we obtain the following relic abundance:
Qoh 2 = 0.09 (T-V)
(3.38)
In order to avoid nucleosynthesis constraints, we need to decay early. This can be
achieved either raising the hierarchy between Higgsino and Gravitino, or raising the
mass of the Higgsino. Since
QLSP
=-mLS5pNLSP,
we can not grow too much with
the hierarchy, and so we are forced to raise the mass of the Higgsino.
This is exactly one of the two directions to go in the parameter space we had
outlined in the first sections, and it is the one which is less favourable for detection
at LHC.
The results of an actual computation are shown in fig.3-8, where we plot the
allowed region for the Higgsino NLSP, in the plane mg,,
quantity
3m
3m = mNLSP - mr.
The
well represents the available energy for decay, and, obviously, can not be
negative. The Hadronic and the Electromagnetic constraints we use come from the
numerical simulations done in [18, 19]. There, constraints are given as upper limit
on the quantity
= B(EM,Had)E(EM,Had)Y as a function of the time of decay. We then
apply this limit to our NLSPs computing both the time of decay and the quantity 5
with the formulas given in the former section.
As we had anticipated, the cosmologically allowed region is given by:
mgr
< 4 x 102 GeV
mf > 20TeV
59
(3.39)
(3.40)
Log(MLsp
/ G eV)
4
1
0
5
5
\\
\-I\.
.I.\~~~~~~~~~~~
I
.I.
Had -:
.I.
. 1.
4
\
M,=5
n TeV
4
I
I
'
I, Xs
r
\' \
>
3 %
3
I sec
\\
---- ------
'0
re
m
---
--
I
<
bD
2
2
2
-lO
EM
10
D-i
1
1
DM
o
0
0
·
·
·
·
1
2
3
4
Log(MLsp/GeV)
Figure 3-8: Constraints for the Higgsino NLSP, Gravitino LSP. There is no allowed
region. The long dashed contour delimitats from above the excluded region by the
hadronic constraints from BBN, the dash-dot-dot contour represents the same for EM
constraints from BBN, the dash-dot lines represent the region within which QDM is
within the experimental limits; finally, we show the short dashed countors where the
Higgsino decays to Gravitino at 1 sec, 105 sec, and 1010sec, and the solid line where
the Higgsino is 5 TeV heavy, which represents the upper bound for Gauge Coupling
Unification. For Neutral Wino and Chargino NLSP, the result is very similar.
This mass range is not allowed by gauge coupling unification, as, for Gravitino LSP,
we have to use the upper bound on NLSP coming from gaugino mass unification
initial conditions. So we conclude that the Higgsino NLSP is excluded. This is a very
nice example of how much we can constraint physics combining particle physics data,
and cosmological observations.
Finally, note how the hadronic constraints raise the limit on the Higgisino mass
60
of approximately one order of magnitude.
In the case of the Neutral Wino NLSP, and Chargino NLSP, there are basically
no relevant differences with respect to the case of the Neutral Higgsino, the main
reason being the fact that the many annihilations channels lead to an high mass for
the NLSP, exactly in parallel to the case of the Higgsino. We avoid showing explicitly
the results, and simply say that, for all of them, the cosmologically allowed parameter
space is very similar in shape and values to the one for Higgsino, with just this slight
correction on the numerical values:
mgr < 5 102 GeV
(3.41)
mWo > 30TeV
(3.42)
for the Wino case. Notice that the mass is a little higher than in the Higgsino case,
as the Wino is naturally more interacting.
In fact, its relic abundance is given by
(again, using a on porpuse modified version of Dark Susy [10] ):
,h = 0.02
wo
(TeV)
(3.43)
For the Chargino, the mass limit is even higher:
mgr < 103GeV
(3.44)
mW+ > 40TeV
(3.45)
All of these regions are excluded by the requirement of gauge coupling unification.
3.5.2
Photino NLSP
As we saw in the former section, hadronic constraints pushed the mass of the NLSP
different from a Bino one above the 10 TeV scale, with the resulting conflict with
gauge coupling unification. Anyway, just looking at the electromagnetic constraints
in fig.3-8 , one can see that a particle that will not decay hadronically at the time
61
of Nucleosynthesis will be allowed to be one order of magnitude lighter. This makes
the photino, A = cos(0w)B + sin(Ow)W3 , a natural candidate to be a NLSP with
Gravitino LSP.
Computing the relic abundance of a Photino is rather complicated, even in Split
Susy. The reason is that co-hannilation channels with the charged Winos makes a
lot of diagrams allowed. In order to estimate the Photino relic abundance, we then
observe that, because of the fact that the Bino is very weakly interacting in Split Susy,
Photino annihilation channels will be dominated by the contribution of the channels
allowed by the Wino component. We then quite reliably estimate the Photino relic
abundance starting from the formula for the relic abundance of a pure Wino particle
we found before:
Q2WinoNLSPh
=
0.02
(3.46)
TeV
and consider that the Photino has a Wino component equal to sin(Gw) So, for the
Photino case we will have:
QPhotinoNLSP
(w)40.02
sini
MTeV
) (M)
2
A
037TeV
(34)
(3.47)
We then obtain the allowed region shown in fig.(3-9). The graph is very similar
to the Higgsino case, with the difference that the Electromagnetic constraints are less
stringent than the Hadronic ones. This allows to have the following region:
700 GeV < MA < 5 TeV
(3.48)
the lightest part of which might be reachable at LHC. However, already if we allow an
hadronic branching ratio of 10- 3 , we see that the Photino NLSP becomes excluded.
So, we conclude that a Photino NLSP is in principle allowed, but only if we fine tune
it to be extremily close to a pure state of Photino.
62
o
Log(MLSplGeV)
2
1
5
I
"
"
"
"
I
"
"
"
"
I
"
"
"
, ,
"
/--
"
"
-- . --
"
"
"
""""
"
"
4
"
o
~ 3
,,'
,,'
i
!
_ .. ' __ .~;,;.;:~
/.---"-"-
~
__ --<,:..''''-:': :'\.,\"\.-
-;#'",»
.
J ,.,,;;;:.,.;;-;,,""",.,"".,
,.
lOsee
2
I
~ ,
-
1010s~_, __.,-.,-
.'- - - - - - - - - - - - -- -- - - - -- -- - - - - -- -- -- - - - - - - -
,-'
CMB
" ....
L
_
2
/\\ ! i
DM
",
"'
\ \
/
1
t --------------. ---- ---- ---
\.\,,
_,:.:;~.,
..;.~ -,;;. ~.;;;.,~
4
I
,,'
..:. - ... - ... - ... -
bO
I
I
EM
::s
--
"
.._--"-
~~
..
---.~~~~.~;~"",.,.,
"<:'><~"~",,
MK=5TeV
10 ~,
.. '
."",
"
"
~
5
I
"
"
.3
4
3
,., ,
! i
! i
!i
-i .-t. - -,
- - -- - - - - - - - - - - -
- -- - - -.
1
. I
! i
! i
! i
! i
! i
! i
! i
~i
o
o
o
3
1
4
Figure 3-9: Shaded is the allowed region for the Photino NLSP, Gravitino LSP. The
long dashed contour delimitates from the left the region excluded by CMB, the dashdot-dot contour delimitates from above the region excluded by the EM constraints
from BBN in the case Bh rv 0, the dashed-dot-dot-dot countor represents the same
for Bh rv 10-3. The region within the dash-dot lines represents the region where nDM
is within the experimental limits; finally, we show the short dashed contours which
represent where the Photino decays to Gravitino at 1 sec, 105 sec, and 1010 sec, and
the solid contour where the Photino is 5 TeV heavy, which represents the upper limit
for Gauge Couplig Unification. We see that already for Bh rv 10-3 a Photino NLSP
is excluded.
3.5.3
Bino NLSP
Bino NLSP is a very good candidate for avoiding all the cosmological constraints.
In Split Susy, a Bino NLSP is almost not interacting.
63
For a pure Bino, the only
interaction which determines its relic abundance is the annihilation to Higgs bosons
through the exchange of an Higgsino. Since this cross section is naturally very small,
by eq.(3.12),
QNLSP
is very big. This means that, in order to create the right amount
of DM (see eq.(3.15)), we need to make the Gravitino very light. And this is exactly
what we need to do in order to avoid the nucleosynthesis bounds. We conclude then
that, of the two directions to solve the DM and the nucleosynthesis problems that
we outlined in the former sections, a Bino NLSP would naturally pick up the one
which is the most favorable for LHC detection. However, as we saw in the section
on Gauge Coupling Unification (see fig.3-4), the Higgsino can not be much heavier
than the Bino. This implies that a Bino like NLSP will have to have some Higgsino
component unless it is very heavy and the off diagonal terms in the mass matrix
are uniportant.
As a consequence, new annihilation channels opens up for the Bino
NLSP through its Higgsino component. This has the effect of diminishing the relic
abundance of a Bino NLSP with respect to the naive thinking we would have done if
we neglected the mixing. As a result, the cosmological constraints begin to play an
important role in the region of the spectrum we are interested in, and, at the end,
considering the upper limit from gauge coupling unification, exclude a Bino NLSP.
In order to compute the Bino NLSP relic abundance, we again modify the Dark
Susy code[10]. The results are shown in fig.3-10. As anticipated, we see that the
relic abundance strongly depends on the ratio beteween the Bino and the Higgsino
masses M1 , . The relic abundance of the Bino NLSP becomes large enough to avoid
the cosmological constaints only for so large values of the ratio between /A and M1
which are not allowed by Gauge Coupling Unification. We conclude, then, that a
Bino NLSP is not allowed.
3.6
Extradimansional LSP
When we consider generic possibilities to break Susy, we can have, further than the
Gravitino, other fermions in a hidden sector which are kept light by an R symmetry.
The implications of these fermions as being the LSP can be quite different with respect
64
to the case of Gravitino LSP, as we will see in this section. Here, we concentrate os
Susy breaking in Extra Doimensions, where, as it was shown in [1], it is very generic
to expect a light fermion in the hidden sector.
In Susy breaking in Extra Dimension, one can break Susy with a radion field,
which gets a VEV. Its fermionic component, the Goldstino, is then eaten by the
Gravitino which becomes massive. Even though at tree level there is no potential,
one sees that at one loop the Casimir Energy makes the radius instable. One can
compensate for this introducing some Bulk Hypermultiplets, finding that, however,
the cosmological constant is negative. Then, in order to cancel this, one finds that he
has to introduce another source of symmetry breaking, a chiral superfield X localized
on the brane (see for example [1]). This represents a sort of minimal set to break
Susy in Extra Dimensions. If one protects the X field interactions with a U(1) charge,
than one finds that the interactions with the SM particles are all suppressed by the
5 dimensional Plank Mass, of the form:
d40 1 XtXQtQ
(3.49)
This induces the following mass spectrum [1]:
1rM3
Mr ;;MSra
mgr N iM
~q2
ms8
where M 2
M5
M5 ;miX,A
mx
M9
5
(3.50)
rM53 are the 4 and 5 dimensional Plank constants, and where Mi are
the gaugino masses.
It is quite natural to use the extradimension to lower the higher dimensional Plank
mass to the GUT scale, a la Horawa-Witten [29], M 5 - MGUT
'
3x
1016
GeV. We
have this range of scales[1]:
mgr
1013GeV; ms - 10 9 GeV; mradion _ 107GeV; M, ,u, m,
-x 100GeV
(3.51)
We notice that we have just reached the typical spectrum of Split Susy, in a very
65
natural way: we break Susy in Extra Dimension, stabilize the moduli, and we introduce a further Susy breaking source to compensate for the cosmological constant.
We further notice that there is no much room to move the higher dimensional Plank
mass M5 away from the Horawa-Witten value. In fact, the fermion mass scales as
so, a slight change of M5 makes the fermions of Split Susy generically either
too heavy, making them excluded by gauge coupling unification, or too light, making
(-),
them conflict with collider bounds.
Concerning the study of the LSP, we notice that the fermionic component of the
X field we have to introduce in order to cancel the cosmological constant is naturally
light, of the order of the mass of the Gauginos. So, it is worth to investigate the case
in which this fermion is the LSP, and how this case differs from the case in which the
LSP is the Gravitino.
Concerning the DM abundance, nothing changes with respect to the case of the
gravitino LSP, so, we can keep the former results.
Next step it is to evaluate the decay time, to check if the nucleosynthesis and
CMB constraints play a role.
To be concrete, let us concentrate on the Higgsino NLSP. When the Higgsino is
heavier than the Higgs, the leading contribution to the decay of the Higgsino will
come from the tree level diagram mediated by the operator[l]:
Jd2
_20m2X
m2 HUHd
(3.52)
128
M
4( m2
The decay time is then given by:
7r
+, mH
'-M5
-
__
mh
\
H/
4mHx
m
In the limit of mh > mrx, mh this expression simplifies to:
T1 8( s(3.54)
(MA
128
7r9
ft
66
M5
-1)2
(3.53)
Estimating with the number we just used before, we get:
T
1014
(Te)
m ff
sec
(3.55)
This time is so long before nucleosynthesis, that all the BBN constraints we found in
the former case for the Gravitino now disappear. Clearly, this statment is not affected
if we vary M5 in the very small window allowed by the restrictions on the fermionic
superpartners'
spectrum. Basically, in this mass regime, the only constraint which
will apply will be the one from
QDM.
As we can see form the formula for the higgsino
relic abundance, the region where Higgsino is lighter that Higgs is not relevant, and
is excluded by the constraint on Dark Matter abundance.
So, we conclude that nucleosynthesis and CMB constraints do not apply in the case
in which the LSP is the field Ox, and the Higgsino is the NLSP, the only constraints
which applies are the one coming from QDM and the one coming from gauge coupling
unification. In fig.3-11, we show the allowed region. While the full region is quite
large, and covers a rather large phase space, there is a region where the Higgsino is
rather light,
--
2 TeV, and the mass of the field Ox is constrained quite precisely to
be around 2 TeV. The region is bounded from above by the limit on gauge coupling
unification at around 18 TeV, as in this case we must allow also for anomaly mediated
initial conditions for Gauginos mass at the intermediate scale. This region is not
extremely attractive for LHC.
For gaugino NLSP, the situation is very similar, as the decay of the gauginos to
the field Ox is mediated by the same kind of operator as for the Higgisino case [1]:
d
2WW
(3.56)
where W is the gaugino vector supermultiplet. Clearly, again in this case, the decay
time will be way before the time of BBN. In this cases, the curves that delimitate
the allowed region are practically identical to the one of the Higgsino, with the only
difference that the region where the NLSP is the lightest and it is practically degenerate with the Ox, correspont to an higher mass of
67
3 TeV, more difficult to see at
LHC.
Similarly occurs for the Bino NLSP, with the only difference that the region allowed by the Dark Matter constraint is a bit different with respect to the case of
Higgsino and Wino. We obtain the allowed region shown in fig.3-12, where we see
that the spectrum is very light, with Bino and Higgsino starting at tens of GeV ,
and gluinos at 200 GeV, with ox in the range 101 - 103 GeV . This is a very good
region for LHC. Notice that the upper limit on the Bino mass is again 5 TeV, as in
the case the Bino is the NLSP, we can not have anomaly mediated initial conditions
for Gaugino mass at the intermediate scale.
3.7
Conclusions
In Split Susy, the only two motivations to expect new physics at the TeV scale are
given by the requirement that gauge coupling unification is achieved, and, mostly,
that the stable LSP makes up the Dark Matter of the Universe. This is true in the
standard scenario where the LSP is a neutralino. Here we have investigated the other
main alternative for the LSP, that is that the LSP is constituted by a hidden sector
particle. A natural candidate for this is the Gravitino, which here we studied quite
in detail. Nevertheless, it is true that among the different possibilities we have in
order to break Susy, one can expect the appearence in the spectrum of another light
fermion protected by R symmetry. Here, as an example, we study the case of a light
fermion arising in Susy breaking in Extra Dimension.
The requirement to obtain gauge coupling unification limits the masses for the
fermions to be less than 5 TeV or 18 TeV, according to the different initial conditions
for the Gaugino masses at the the intemediate scale i.
In this range of masses, we have seen how constraints from Nucleosynthesis put
strong limits on the allowed region. In fact, there are two competing effects: in
order to avoid Nuclosynthesis constraints, the NLSP must decay to the LSP early,
and this is achieved creating a big hierarchy between the NLSP and the LSP. On
the other hand, this hierarchy tends to diminish the produced
68
QLSP,
and in order to
compensate for it, the NLSP tends to be heavy. This goes against the constraints
from gauge coupling unification. This explains why a large fraction of the parameter
space is excluded.
The details depend on the particular LSP and NLSP.
3.7.1
Gravitino LSP
Gravitino LSP forces us to consider Gaugino Mass Unification at the intermediate
scale as initial condition. This implies that we have to live with the more restrictive
upper limit on fermionic masses from gauge coupling unification: 5 TeV.
At the same time, the typical decay time of an NLSP to the Gravitino is at around
1 sec, and this goes exactly into the region where constraints from Nucleosynthesis
on ElectroMagnetic and Hadronic decays apply.
The final result is that only if the NLSP is very pure Photino like, then the
Gravitino can be the LSP, with a Photino between 700 GeV and 5 TeV. If the NLSP
is different by this case, then the Gravitino can not be the LSP. The reason is that a
very Photino like NLSP can avoid the constraints on BBN on hadronic decays, which
are much more stringent than the ones coming from electromagnetic decays, and so it
can be light enough to avoid the upper limit on its mass coming from gauge coupling
unification.
This is very good news for the detactability of Split Susy at LHC. In fact, if the
Gravitino was the LSP, than the NLSP could have been much heavier than around 1
TeV, making detection very difficult. In the study in this chapter we show that this
possibility is almost excluded.
3.7.2
ExtraDimensional LSP
Following the general consideration that in breaking Susy we might expect to have
some fermion other than the Gravitino in the hidden sector which is kept light by an
R symmetry, we have studied also the possibility that the fermionic component of a
chiral field which naturally arises in Extra Dimensional Susy breaking is the LSP. In
69
this case the time of decay is so early that no Nucleosynthesis bounds apply, so, the
only constraints applying are those from Dark Matter and gauge coupling unification.
Concerning gauge coupling unification, in this case we must consider the possibility
that the Gaugino Mass initial conditions are also those from anomaly mediation. This
implies that we have to give the upper limit of 18 TeV to Fermions' mass.
Having said this, the lower bound on the mass is given by the DM constraint.
In fact, it is clear that
QNLSP
must be greater than QLSP. As a consequence, the
Mass of the NLSP has to be greater than the one found in [2, 3] for the case in which
these particles where the LSP. As a consequence, Charginos and Neutralinos NLSP
are in general allowed, but they are restricted to be heavier than 1 TeV. It is quite
interesting that, in this cases, the LSP is restricted to be in the range 100-1000 GeV.
Again, there is an exception: the Bino. In this case, the LSP and NLSP are much
lighter than in the case of the others Neutralinos NLSP, with an NLSP as light as a
few decades of GeV.
70
Log(M
LSP
JOeV)
-5
-4
-3
-2
-1
o
1
2
3
-5
-4
-3
-2
-1
0
1
2
3
Log(M
LSP
JOeV)
Figure 3-10: Constraints for Bino NLSP. The long dashed countour delimitates from
the left the excluded region by the hadronic constraint from BBN, the dash-dotdot contour represents the same for EM constraints from BBN, the dash-dot lines
represents the ratio between the Higgsino mass and the Bino mass necessary for
f2LSP to be equal to observed DM amount; the solid line represents the upper limit
from GUT, while the dash-dot-dot-dot line represents the lower limit from LEP; we
also show in short dashed the contours where the Bino decays to Gravitino at 1 see,
105 see, and 1010 see, and in dotted some characterstic contours for the Bino mass.
CMB constraint plays no role here. We take M2 ~ 2M1, as inferred from gaugino
mass unification at the GUT scale [2], and we see that no allowed region is present.
71
Log(MLSP JGeV)
1
5
3
2
""
4
5
"
............
...... ......
...... ............
"
......
"
............
"
............ ......
Mn=18TeV
4
4
b
OQ
i:
3~
CI.l
"'0
I
a::
t"""
CI.l
/
2
)<
'-"
DM
1
1
o
o
1
2
Log(MLSP JGeV)
3
4
Figure 3-11: Shaded is the allowed region for the Neutral Higgsino NLSP, 'l/Jx LSP.
Since there are no constraints from CMB and BBN, the only constraints come from
nDM, which delimitates the region within the dash-dot lines, and Gauge Coupling
Unification, which set the upper bound of 18 TeV with the solid line. For Neutral
Wino and Chargino NLSP the result is very similar.
72
Log(M
/GeV)
LSP
-5
-4
-3
-2
-1
0
1
2
3
5 \
5
\'
\ \\_\
'\\ \ . ,x,~]%A
.
IA
4 A
AY-.R
\
4
S
C,
3
3
zEn
0o
2
2
2
1
Z
C
1
0
0
-5
-4
-3
-2
-1
Log(M
0
1
2
3
/GeV)
LSP
Figure 3-12: Shaded is the allowed region for the Bino NLSP, 'x LSP. The dash-dot
lines represent the ratio between the Higgsino mass and the Bino mass in order for
QLSP to be equal the observed amount of DM. The dotted lines are some characteristic
countors for the Bino mass. The solid line is the upper limit from GUT, while the
dash-dot-dot-dot line is the lower limit from LEP. We take M2 - 2M1 .
73
Bibliography
[1] N. Arkani-Hamed and S. Dimopoulos, arXiv:hep-th/0405159.
[2] G. F. Giudice and A. Romanino, arXiv:hep-ph/0406088.
[3] A. Pierce, arXiv:hep-ph/0406144.
[4] A. Masiero, S. Profumo and P. Ullio, arXiv:hep-ph/0412058.
[5] M. A. Luty and N. Okada, JHEP 0304, 050 (2003) [arXiv:hep-th/0209178].
[6] N. Arkani-Hamed, S. Dimopoulos, G. F. Giudice and A. Romanino, arXiv:hepph/0409232.
[7] J. Bernstein,
L. S. Brown and G. Feinberg, Phys. Rev. D 32 (1985) 3261.
[8] R. J. Scherrer and M. S. Turner, Phys. Rev. D 33 (1986) 1585 [Erratum-ibid.
D
34 (1986) 3263].
[9] See for example, Kolb and Turner, The Early Universe, Perseus Publishing, 1990
[10] P. Gondolo,
J. Edsjo, P. Ullio, L. Bergstrom,
M. Schelke and E. A. Baltz,
arXiv:astro-ph/0211238.
[11] S. Eidelman et al. [Particle Data Group Collaboration], Phys. Lett. B 592 (2004)
1.
[12] G. Altarelli and M. W. Grunewald, arXiv:hep-ph/0404165.
[13] S. Aoki et al. [JLQCD Collaboration],
Phys. Rev. D 62 (2000) 014506 [arXiv:hep-
lat/9911026].
74
[14] Y. Suzuki et al. [TITAND Working Group Collaboration], arXiv:hep-ex/0110005.
[15] C. L. Bennett et al., Astrophys. J. Suppl. 148 (2003) 1 [arXiv:astro-ph/0302207].
[16] D. N. Spergel et al. [WMAP Collaboration], Astrophys. J. Suppl. 148 (2003) 175
[arXiv:astro-ph/0302209].
[17] R. H. Cyburt, J. R. Ellis, B. D. Fields and K. A. Olive, Phys. Rev. D 67, 103521
(2003) [arXiv:astro-ph/0211258].
[18] M. Kawasaki, K. Kohri and T. Moroi, arXiv:astro-ph/0402490.
[19] M. Kawasaki, K. Kohri and T. Moroi, arXiv:astro-ph/0408426.
[20] K. Jedamzik, Phys. Rev. D 70 (2004) 083510 [arXiv:astro-ph/0405583].
[21] K. Jedamzik, Phys. Rev. D 70 (2004) 063524 [arXiv:astro-ph/0402344].
[22] J. Fixsen, E. S. Cheng, J. M. Gales, J. C. Mather, R. A. Shafer and E. L. Wright,
Astrophys. J. 473 (1996) 576 [arXiv:astro-ph/9605054].
[23] K. Hagiwara et al. [Particle Data Group Collaboration], Phys. Rev. D 66 (2002)
010001.
[24] W. Hu and J. Silk, Phys. Rev. Lett. 70 (1993) 2661.
[25] W. Hu and J. Silk, Phys. Rev. D 48 (1993) 485.
[26] J. L. Feng, S. Su and F. Takayama, arXiv:hep-ph/0404231.
[27] J. L. Feng, A. Rajaraman and F. Takayama, Phys. Rev. Lett. 91 (2003) 011302
[arXiv:hep-ph/0302215].
[28] J. L. Feng, A. Rajaraman and F. Takayama, Phys. Rev. D 68 (2003) 063504
[arXiv:hep-ph/0306024].
[29] P. Horava and E. Witten, Nucl. Phys. B 460 (1996) 506 [arXiv:hep-th/9510209].
75
Chapter 4
Hierarchy from Baryogenesis
We study a recently proposed mechanism to solve the hierarchy problem in the context
of the landscape, where the solution of the hierarchy problem is connected to the
requirement of having baryons in our universe via Electroweak Baryogenesis. The
phase transition is triggered by the fermion condensation of a new gauge sector which
becomes strong at a scale A determined by dimensional transmutation,
and it is
mediated to the standard model by a new singlet field. In a "friendly" neighborhood
of the landscape, where only the relevant operators are "scanned" among the vacua,
baryogenesis is effective only if the higgs mass mh is comparable to this low scale A,
forcing mh - A, and solving the hierarchy problem. A new CP violating phase is
needed coupling the new singlet and the higgs field to new matter fields. We study
the constraints on this model given by baryogenesis and by the electron electric dipole
moment (EDM), and we briefly comment on gauge coupling unification and on dark
matter relic abundance. We find that next generation experiments on the EDM will
be sensitive to essentially the entire viable region of the parameter space, so that
absence of a signal would effectively rule out the model.
76
4.1
Introduction
The most problematic characteristic of the Standard Model (SM) seems to be the
smallness of its superrenormalizable couplings, the higgs mass m', and the cosmological constant A, with respect to the apparent cutoff of the theory, the Planck
mass, Mpl. These give rise respectively to the hierarchy, and cosmological constant
problems. According to the naturalness hypothesis, the smallness of these operators
should be understood in terms of some dynamical mechanism, and this has been a
driving motivation in the last two decades in the high energy theory community.
In the last few years, several things have changed.
Concerning the hierarchy problem, experimental investigation has shown that already many of the theoretically most attractive possibilities for the stability of the
weak scale, such as supersymmetry, begins to be disfavored, or present at least some
fine tuning issues [1, 2].
Concerning the cosmological constant problem, there was the hope that some
symmetry connected to quantum gravity would have forced the cosmological constant
to be zero. However, first, cosmological observations have shown evidence for a non
zero cosmological constant in our present universe [3, 4]; second, from the string theory
point of view, two main things have occurred: on the one hand, consistent solutions
with a non zero cosmological constant have been found [5, 6, 7, 8, 9], and, on the other
hand, it has been becoming more and more clear that string theory appear to have a
huge landscape of vacua, each one with different low energy parameters [10, 12].
If the landscape is revealed to be real, it would force a big change in the way
physics has to be done, and some deep questions may find a complete new answer.
In particular, it is conceivable that some characteristics of our low energy theory
are just accidental consequences of the vacua in which we happen to be. This is a
very big step from the kind of physics we are used to, and its consequences have
been explored recently in [11]. There , it is shown that the presence of a landscape
offers a new set of tools to address old questions regarding the low energy effective
theory of our universe. On one hand, there are statistical arguments, according to
77
which we might explore the statistically favored characteristics for a vacuum in the
landscape
[12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22].
On the other
hand,
there
are
also some selection rules due to anthropic reasoning, which we might have to impose
on the vacuum, in order for an observer to be present in it. Pioneering in this, is
Weinberg's "structure principle" [23], which predicts the right order of magnitude for
the cosmologicalconstant starting from the requirement that structures should have
had time to form in our universe in order for life to be present in it.
At this point, it is necessary to speak about predictivity for theories formulated
with a lot of different vacua, as it occurs in the landscape. There are two important
points that greatly affect the predictivity of the theory. First, we must know how the
characteristic of the low energy theory change as we scan among the vacua. Second,
we must know also how the probability of populate such a vacuum changes among the
vacua, also considering the influence of the cosmological evolution. These are very
deep questions whose answer in general requires a full knowledge of the UV theory, and
we do not address them here. However, there is still something we can do. In fact, as
it was pointed out in [11],theories with a large number of vacua can be described in an
effective field theory approach. In such a study, it was shown that it is very natural
for parameters not to effectively scan in the landscape, unless their average value
in the landscape is null. So, it is reasonable to assume that some parameters in the
landscape do scan, and some do not. Among the parameters of the theory, the relevant
operators have a particular strong impact on the infrared properties of the theory.
But exactly because of this, if they are the only ones to scan, their value can be fixed
by environmental arguments. This is however true only if the marginal couplings do
not scan. For this reason, a neighborhood of the landscape in which only the relevant
parameters are scanning, is called a "friendly neighborhood", because it allows to fix
the value of the relevant couplings with environmental reasoning, while the marginal
parameters do not scan, and so they are analyzed with the usual instruments of
physics, judging them on the basis of their simplicity and of their correlations. We do
not know if the true string theory landscape have these properties. However, there is
some phenomenological motivation to expect that this could well be. In fact, in the
78
physics connected to the standard model, we have been able to successfully address
question concerning the marginal parameters with dynamical reasoning, deducing
some striking features as gauge coupling unification, chiral symmetry breaking, or
the weakly interacting dark matter, while, on the contrary, we have been having big
troubles concerning the problems connected to the relevant coupling in the standard
model, the cosmological constant, and the higgs scale, both of which arise large fine
tuning issues. In this chapter, we want to address the hierarchy problem of the
electroweak scale assuming we are living in such a "friendly neighborhood" of the
landscape.
Because of this, we concentrate a little more on the predictivity of a
phenomenological model based on this. As it is clear, predictivity in the landscape is
very much enhanced if there is an infrared fragile feature [11] which must be realized
in the universe in order for it not to be lethal, and which is however difficult to realize
in the scanned landscape. Exactly the fact that a necessary fragile feature is difficult
to realize gives a lot of constraints on the possible vacua in which we should be, or, in
other words, on the parameters of the theory, making then the model predictive. An
example of a fragile feature is the presence of structures in the universe. According
to Weinberg's argument [23], if we require the fact that in the universe there should
be structures, than the value of the cosmological constant is very much tighten to a
value close to the observed one. The presence of structures can then also be used to
anthropically require the lightness of the higgs field in the case dark matter particles
receive mass from electroweak symmetry breaking [11, 24].
In this chapter, we concentrate on another fragile feature which we consider necessary for the development of any sort of possible life in the universe, and which is a
necessary completion on Weinberg's structure principle. In the absence of baryons,
the dark matter would just form virilized structures, and not clumped structures,
which instead are necessary for the development of life. We can construct a model
where baryons are explicitly a fragile feature. Then, we can connect the solutions
of the problem of baryons to the hierarchy problem with a simple mechanism which
imposes a low energy electroweak scale in order for baryons to be present in the universe. This is the line of thought implemented in [11], which is the model on which we
79
focus in this chapter. As we will explain in detail in the next section, the mechanism
is based on the fact that baryogenesis is possible in the model only if the electroweak
scale is close to a hierarchical small scale. The hierarchical small scale is naturally
introduced setting it equal to the scale at which a new gauge sector of the theory
becomes strong through dimensional transmutation.
This mechanism naturally pro-
vides a small scale. In order to successfully implement baryogenesis, we will also need
to add some more CP violation. This will force the presence of new particles and
new couplings. The model becomes also predictive: the lightest of these particle will
naturally be a weakly interacting particle at the weak scale, and so it will be a very
good candidate for dark matter; further, the new CP violating terms will lead to the
prediction of a strong electron electric dipole.
The main purpose of the study in this chapter is to investigate in detail the
mechanism in which baryogenesis occurs in this model, and the consequent predictions
of the model. In particular, we will find that next generation experiments of the
electron Electric Dipole Moment (EDM), together with the turning on of LHC, are
going to explore the entire viable region of the parameter space, constraining it to
such a peculiar region of the parameter space, so that absence of a signal would result
in ruling out the model.
The chapter is organized as follows: in sec. 5.2, we explain in detail the model;
in sec. 5.3 we study the amount of baryons produced; in sec. 5.4 we determine the
electron EDM; in sec. 5.5 we briefly comment on Dark Matter and Gauge Coupling
Unification ; and in sec. 5.6, we draw our conclusions.
4.2
Hierarchy from Baryogenesis
In this section, following [11], we show how we can connect the electroweak scale to
a scale exponentially smaller than the cutoff.
As the presence of baryons is a necessary condition for the formation of clumped
structures, it is naturally to require that the vacuum in which we are should allow
for the formation of a net baryon number in the universe.
80
This is not a trivial
requirement. In fact, in the early hot phase of the universe, before the electroweak
phase transition, baryon number violating interactions through weak sphalerons were
unsuppressed, with a rate given approximately by [25]:
F,, = 6ka5T
where k -
(4.1)
20, with the consequence of erasing all precedently generated baryon
asymmetry (if no other macroscopic charges are present). However, at the electroweak
phase transition, all the Sakharov's necessary conditions [26] for generating a baryon
asymmetry are satisfied, and so it is possible in principle to generate a net baryon
number at the electroweakphase transition, through a processknown in the literature
as electroweak baryogenesis (see [27, 28] for two nice reviews). However, it is known
that the SM electroweak phase transition can not, if unmodified, generate the right
amount of baryons for two separate reasons. On one hand, CP violating interactions
are insufficient [29], and, on the other hand, the phase transition is not enough first
order for preventing weak sphalerons to be active also after the phase transition [30].
Since this last point is very important for our discussion, let us review it in detail.
Let us suppose in some early phase of the universe we have produced some initial
baryon number B and lepton number L, and no other macroscopic charge. In partic-
ular B - L = 0. The quantum number B + L is anomalous, and the equation for the
abundance of particles carrying B + L charge is given by:
d(8 .)
dtwhere
nB+L
=
r (nB+L)
(4.2)
is the number of baryons and leptons per unit volume, s is the entropy
density - T3 , and F is related to the sphaleron rate [27],r ,1
N Te()(),
where
v(T) is the temperature dependent higgs vev (v(To) = 246 GeV as measured in our
vacuum), Nf is the number of fermionic families, and g92is the SU(2) weak coupling.
The reason why reactions destroying baryons are faster than reactions creating them
81
is due to the fact that the relative reaction rate goes as:
+
-Af
IL-
(4.3)
where Af is the difference in free energy, and F± are the sphaleron rates in the
two directions. Now, it is easy to see that the free energy grows with the chemical
potential
AB,
which then grows with the number of baryons nB. In the limit of small
difference, we then get eq.(4.2).
This differential equation can be integrated to give:
(n
(nB+L
B+L )
- /
final
S
Nfi v2mpIle
(
initial
(4_
v(Tc)
)
g, v(Tc)
(4.4)
)
where Tc is the critical density, and where g, is the number of effective degrees of
freedom (
55).
If in the universe there is no macroscopic lepton number, than clearly at present
time we would have no baryons left, unless:
Nf
g*
(mpl e-(
92 v(Tc) <
(T,)
1
(4.5)
which roughly implies the constraint:
v(T) > 1
Tc
(4.6)
This is the so called "baryons wash out", and the origin of the requirement that
the electroweak phase transition should be strongly first order.
Note that this is the same condition we would get if we required sphaleron interactions not to be in thermal equilibrium at the end of the phase transition:
F
<H
82
(4.7)
implies roughly:
4 v(Tc)
72
T, e 92S
T
<
2
c
(4.8)
and so
v(T¢) > 1
(4.9)
Also note that the requirement for the sphalerons not to be in thermal equilibrium
already just after the phase transition is necessary, as otherwise we would have a
baryon symmetric universe, which leads to a far too small residual relic density of
baryons.
Later on, when we shall study electroweak baryogenesis, we shall get a number for
the baryon number. In order to get then the baryon number at, let us say, Big Bang
Nucleosynthesis (BBN), we need to multiply that number by the factor in eq.(4.4).
In the next sections, we shall consider the requirement in eq.(4.6) fulfilled, and we
will consider completely negligible the wash out from the sphalerons coming from
eq. (4.4).
Now, let us go back to the higgs phase transition, and let us assume that in the
neighborhood of the landscape in which we are, all the high energy mechanisms for
producing baryons have been shut down. This is easy to imagine if, for example,
the reheating temperature is smaller than the GUT scale. We are then left with the
only mechanism of electroweak baryogenesis. From the former discussion, it appears
clear that there should be some physics beyond the SM to help to make the phase
transition strong enough.
We may achieve this by coupling the higgs field to a singlet S, with the potential
equal to:
V = AS4 + Ah(hthwhere we have assumed a symmetry S T,
JFC,
S2)2 + mS2 + mhth
(4.10)
-S. We can couple this field to two fermions
which are charged under a non-Abelian gauge group through the interaction
ksSlqJc
83
(4.11)
In order to preserve the symmetry S -+ -S, we give the fields i, oCcharge i. We can
then assume that this sector undergos confinement and chiral symmetry breaking at
its QCD scale determined by dimensional transmutation
< tcV >- A3
(4.12)
which is naturally exponentially smaller than the cutoff of the theory. We assume
that this phase transition is first order, so that departure from thermal equilibrium
is guaranteed.
Now, following our discussion in the introduction, suppose that, scanning in the
landscape, the only parameters which are effectively scanned are the relevant couplings mh and ms. If then we must have baryons in our universe so that clumped
structures can form, than we need to be in the vacuum in which these two parameters
allow for a strong enough first order phase transition in the electroweak sector. So,
we have to require that this phase transition triggers the electroweak phase transition. It is clear that this can only be if it triggers a phase transition in the S field,
which is possible then only if ms is of the order of A. Finally, the phase transition
in S can trigger a strong first order phase transition in the higgs field only if again
mh - ms - A (for a more detailed discussion, see next subsection). So, summarizing,
we see that, the requirement of having baryons in the universe forces the higgs mass
to be exponentially smaller than the cutoff, solving in this way the hierarchy problem.
In order to produce baryons, we still need to improve the CP violating interactions,
that in the SM are not strong enough. We can minimally extend the introduced model
to include a singlet s and 2 SU(2) doublets 9I±, with hypercharge ±1/2 (notice that
they have the same quantum numbers as higgsinos in the Minimal Supersymmetric
Standard Model (MSSM)), with the following Yukawa couplings:
kSss+ k'SF+'_ +ght xJ+.s
+g'h'_s
84
(4.13)
There is then a reparametrization invariant CP violating phase:
0 = arg(kk'g*g'*)
(4.14)
The mass terms for this new fields can be prohibited giving proper charges to the
fields s, 'I±. This implies that, since at the electroweak phase transition these new
fermions get a mass of order of the electroweak scale, and also because of the fact
that the lightest fermion is stable, we actually have a nice candidate for dark matter.
This last point is a connection between the higgs mass, which, up to this point, we
have just assured to be exponentially smaller than the cutoff, and the weak scale,
but this connection will come out quite naturally later. So, if this model happens to
describe our universe, what we should see at LHC should be the higgs, the two new
singlets S and s, and the two new doublets 4'±.
Since the model is particularly minimal, it is interesting to explore the possibility
for generating the baryon number of the universe in more detail. Before doing so,
however, let us see in more detail how the vevs of the higgs and of the singlet S are
changed by the phase transition.
4.2.1
Phase Transition more in detail
Before going on, here we show more in detail how the requirement of having a strong
first order phase transition leads to have ms
-
mh
A.
In unitary gauge, the equations to minimize the potential are (from here on, we
mean by S also the vev of the field S; the meaning will be clear by the contest):
2
4AS3 -
4Ah2S(2
-S
2)
-AS
2)
where T is the critical temperature.
+ m2S + ksA 3E(T, - T) = 0
+ 2m2h
=
0
(4.15)
(4.16)
The first equation comes from the derivative
with respect to S, and we will refer to it as S equation, while for the other we will
use the name h equation.
85
We first consider the case
> 0 and 0
m
> . Before the phase transition, we
have the minimum at the symmetric vacua:
S = O, v = 0, for T> Tc
(4.17)
For T < T, the minimum conditions change, and we can not solve them analytically.
We can nevertheless draw some important conclusions. Let us consider the minimum
equation for S, which is the only equation which changes. Let us first consider the
case ms
>
A. In this case, we can consistently neglect the cubic terms in the S
equation, to get:
S
ksA 3
-2AA 2)
(m
(4.18)
Then, if m2 > 2AAhv2 , we have:
S- -
ksA 3
2
(4.19)
ms
v= 0
(4.20)
while the other solutions are still unphysical, as:
v2 _mh
2
+ kA
<0
(4.21)
ms
Ah
If m << 2AAhv2 , then:
S-
2Aa
(4.22)
2XAhv2
For the higgs, the non null solution is:
v2
h
Ah
,
(
A)
which has some relevant effect for the electroweak phase transition only if
But in that case v
(4.23)
4A2A2v4a
mh
-
A.
A, and from mn<2< 2AAhv 2 we get that m2 <<A 2 , in contradiction
with our initial assumption.
Then, if mS2hv
86
< 1, in the S equation, we can
consistently consider just the cubic term, to get:
1 /3
41/3((A+ AAh)
(4.24)
then, the non null solution for v 2 becomes:
v2
2/3AA2
2
(42/3( + AAh)(A+
Ah
Ah))
(4.25)
which has some relevant effect only if m~h< A2 . However, in this case the condition
Mr-2AAhv
s2
2
<1
would imply
M
< 1, again in contradiction with our assumptions.
So, in order to have some effect on the higgs phase transition, we are left with the
only possibility of having ms
A.
Restricting to this, consistently, we can neglect the linear term in the S equation,
to have:
4AS3 -4AhS ( 2-AS2)
2Ah
AS
=-ksA
3
= -2mh2
(4.26)
(4.27)
which implies the following equation for S:
4AS3 + 4Am2S = -ksA
3
(4.28)
For mh > A, we have:
ksA 3
S=
4Am2
(4.29)
and the non null solution for v is still not physical:
v2
mh +
sA
2<
(4.30)
So, finally, we have that in order to have a strong first order phase transition triggered
by the new sector, we are forced to have mh < A, which is what we wanted to show.
In detail, for mh < A, we implement the following phase transition
87
S:
0
S
-
(4.31)
(4X)1/3
11/21/3
v:
0
21/3
A
..
A1/3
Now, the final step is to show that, not only mh < A and ms < A, but that
actually mh - A and ms
A. The argument for this is that, scanning in the
-
landscape, it will be generically much more difficult to encounter light scalar masses,
as they are fine tuned. So, in the anthropically allowed range, our parameters shall
most probably be in the upper part of the allowed range. So, we conclude that this
model predicts mh
-
ms
A, as we wanted to show.
This phase transition must satisfy the requirement that the sphalerons are ineffective
([c)
> 1), after the phase transition, which occurs at, roughly, the critical
temperature T - A. So:
f1/ 2 (2ks)1 /3
v(T)
11/3
-
T,
1
(4.32)
"d
which can clearly be satisfied for some choices of the couplings. In order to better
understand the natural values of the ratio v(Tc)/Tc in terms of the scalar couplings, it
is worth to notice that the coupling A appears in the lagrangian as always multiplied
by
Ah.
So, the coupling which naturally tends to be equal to the other ones in the
lagrangian is A - AhA. With this redefinition, we get the constraint:
v(T)
T,
(A )1/
2
Ah
(2
1
A
r1>
(4.33)
(4.33)
which can clearly be satisfied with some choice of the parameters. It is also worth to
write the ratio of the vevs of S and h after the phase transition:
-v
(=A )(4.34)
"
We can also analyze the case in which mh < 0. In this case, the electroweak phase
transition would occur before the actual strong sector phase transition.
88
However,
since we know that in the SM the phase transition is neither enough first order, nor
enough CP violating, we still need, in order to have baryon formation, to require
that the strong sector phase transition triggers a phase transition in the higgs sector.
It is then easy to see that all the former discussion still applies with tiny changes,
and we get the same condition m~2,,
m2
A2 . There is a further check to make,
though, which is due to the fact that, in order for baryogenesis to occur, we need
an unsuppressed sphaleron rate in the exterior of the bubble. This translates in the
requirement, for m2 < 0:
mh
lnh<
1
A
(4.35)
which can be satisfied in some vacua.
In the next of this chapter, we will always assume that these conditions are satisfied, and the sphalerons are suppressed in the broken phase.
Now, we are ready to treat baryogenesis in detail.
4.3
Electroweak Baryogenesis
During the electroweak phase transition, we have all the necessary conditions to fulfill
baryogenesis [27, 28]. We have departure from thermal equilibrium because of the
phase transition; we have CP and C violation because of the CP and C violating interactions; and finally we have baryon violation because of the unsuppressed sphaleron
rate. The sphaleron rate per unit time per unit volume is in fact unsuppressed in the
unbroken phase (see eq.(4.1)).
There are various different effects that contribute to the final production of baryons.
For example, CP violation can be due just to some CP violating Yukawa coupling
in the mass matrix, or it can be mainly due to the fact that, in the presence of the
wall, the mass matrix is diagonalized with space-time dependent rotation matrixes,
which induce CP violation. Further, CP violation can be accounted for by some time
dependent effective coupling in some interaction terms. Baryon number production
as well can be treated differently. In the contest of electroweak baryogenesis which we
are dealing with, there are mainly two ways of approaching the problem, one in which
89
the baryon production occurs locally where the CP violation is taking place, so called
local baryogenesis, or one in which it occurs well in the exterior of the bubble, in the
unbroken phase, in the so called non local baryogenesis. At the current status of the
art, it appears that the non local baryogenesis is the dominant effect, at least for not
very large velocity of the wall vw, which however is believed to be not large because
the interactions with the plasma tend to slow down the wall considerably [31], and
we concentrate on this case (see Appendix A for a brief treatment of baryogenesis in
the fast wall approximation).
In order to compute the produced baryon abundance, we follow a semiclassical
method developed in [32]. A method based on the quantum Boltzmann equation
and the closed time path integral formalism was developed in [33], making a part of
the method more precise. However, the corrections given by applying this procedure
to our case can be expected to be in general not very important once compared
to the uncertainties associated to our poor knowledge of certain parameters of the
electroweak phase transition, as it will become clear later, which make the computed
final baryon abundance reliable only to approximately one order of magnitude [32].
Further, it is nice to note that our general conclusions will be quite robust under
our estimated uncertainty in the computation of the baryon abundance. For these
reasons, the method in [32] represents for us the right mixture between accuracy and
simplicity which is in the scope of our study.
Since the method is quite contorted, let us see immediately where the basic ingredients for baryogenesis are. Departure from thermal equilibrium obviously occurs
because of the crossing of the wall. C violation occurs because of the V - A nature
of the interactions. CP violation occurs because the CP violating phases in the mass
matrix are rotated away at two different points by two different unitary matrix such
that U(x)tU(x + dx) =! 1. There could be other sources of CP violation, but, in
our case, this is the dominant one. Baryon production occurs instead well in the
exterior of the wall, in the so called non-local baryogenesis approach, where the weak
sphaleron rate is unsuppressed.
Let us anticipate the general logic. The calculation is naturally split into two
90
part. In the first part, we compute the sources for CP violating charges which are
due to the CP violating interactions of the particles with the incoming wall. This
calculation will be done restricting ourself to the vicinity of the wall, and solving a
set of coupled Dirac-Majorana equations to determine the transmission and reflection
coefficients of the particles in the thermal bath when they hit the wall, which are
different for particles and antiparticles. In the wall rest frame, the wall is perceived
as a space-time dependent mass term. This will give rise to a CP violating current.
The non null divergence of this current will be the input of the second part of the
calculation.
In this second part, we shall move to a larger scale, and describe the
plasma in a fluid approximation, where we shall study effective diffusion equations.
The key observation is that, once the charges have diffused in the unbroken phase,
thermal equilibrium of the sphalerons will force a net baryon number. In fact, in the
presence of SU(2) not neutral charges, the equilibrium value of the baryon number is
not zero:
(B + L)eq = E cQi
(4.36)
where ci are coefficients which depends on the different charges . Once produced, the
baryon number will diffuse back in the broken phase, where, due to the suppression
of the sphaleron rate, it will be practically conserved up to the present epoch. This
will end our calculation.
In the next two subsections, we proceed to the two parts of the outlined calcula-
tion.
4.3.1
CP violation sources
We begin by computing the source for the CP violating charges, following [32]. We
restrict to the region very close the the wall, so that the wall can be considered flat,
and we can approximately consider the problem as one dimensional. We consider a
set of particles with mass matrix M(z) where z is the coordinate transverse to the
wall, moving, in the rest frame of the wall, with energy-momentum E, k. Taken z 0
as the last scattering point, these particles will propagate freely for a mean free time
91
i, when they will rescatter at the point zo + Tri, where v is the velocity perpendicular
to the wall, ktr/E. Now, because of the space dependent CP violating mass matrix,
these particles will effectively scatter, and the probability of being transmitted and
reflected will be different for particles and antiparticles. This will create a current for
some charges, whose divergence will then be the source term in the diffusion equations
we shall deal with in the next section. The effect will be particularly large for charges
which are explicitly violated by the presence of the mass matrix, and we shall restrict
to them.
We introduce J as the average current resulting from particles moving towards
the positive and the negative z direction, between zo0 and z0o + A, where A =
T
TV,
and
is the coherence time of the particles due to interactions with the plasma [32]. J
are the CP violating currents associated with each layer of thickness A. J+ receives,
for example, contributions from particles originating from the thermal bath at z0 with
velocity v, and propagating until z0o+ A, as well as from particles originating at zo+ A
with velocity -v, and being reflected back at zo0 + A. The formula for J+ is given by:
J+ = (
(z
( (TtQT - TtQT)) Tr (Pzo+ (tQR
J_= (Tr (Pzo(RQR-
-
RtQ)) - Tr (Pzo+A(TtQT
RtQR))) (1,0,0, ) (4.37)
tQT))) (1,0,0, -v)
(4.38)
where R(R) and T(T) are reflection and transmission matrices of particles (antiparticles) produced at zo with probability pzo, evolving towards positive z; while T and
R are the correspondent quantities for particles produced at z0 + A with probability
pzo+ and evolving towards negative z; v is the group velocity perpendicular to the
wall at the point zo,
0 and
is the same quantity at zo + A; Q is the operator corre-
spondent to the chosen charge; and the trace is taken over all the relevant degrees of
freedom and averaged over location z0o within a layer of thickness A. When boosted
in the plasma rest frame, these currents will become the building block to construct
the CP violating source for the charges that, diffusing into the unbroken phase, will
let the production of baryon number possible.
92
Now, consider a small volume of the plasma, in the plasma rest frame. As the
wall crosses it, it leaves a current density equal to (J+ + J-)pIasmaevery time interval
T,
where the subscript plasma refers to the quantity boosted in the plasma frame.
So, at a time t, the total current density accumulated will be given by:
t') + J(X t )))ilasma
dt'-(J(I
S =
TR
(4.39)
7
where TR is the relaxation time due to plasma interaction. From this, the rate of
change of the charge Q per unit time is given by:
VyQ(,t) =
O,s"=
~~~~~~~~-1
1
(J+(Zt) + J-_(,t))plasma0 (J+(, t --TR)+ J (t
(4.40)
- 7-R))plasma
-7
-j/
'TR
dz&z(J + J- )plasma
Since in the SM CP violation is very small, and already proved to be not enough
to account for the baryon number of the universe, we can clearly concentrate in the
sector of the neutral particles of the new theory, where the mass matrix is given by:
kS(z)
M(z)=
(
0o
k'S(z)
I
k'S(z)
O
The transmission and reflection coefficients can be found by solving the free coupled
Dirac-Majorana equations for these particles with the mass matrix given in M(z).
We can solve this by a method developed in [32], in a perturbative expansion in mass
insertion. As it is explained in [32, 34], the small parameter in the expansion is mA,
where m is the typical mass of the particles, in the case wA < 1, or m/w in the case
wA > 1. In both cases, the expansion parameter is smaller than one. This can be
understood noticing the analogy of our system with the scattering off a diffracting
medium with a step potential of order m. In that case, reflection and transmission
are comparable (and this is the only case in which we produce a net CP violating
93
charge) only if the wave packet penetrates coherently over a distance of order 1/m,
and has few oscillation over that distance. Suppression of the reflection occurs both if
mA < 1 and if m/w < 1. In the first case, this is because only a layer of thickness A
contributes to the coherent reconstruction of the reflected wave, while in the second
case, because fast oscillations tend to attenuate the reconstruction of the reflected
wave. Up to sixth order in the mass insertion, we get:
T =
dzl
1-
dzl
+|
J
j
dz2 M 2 M*e 2iw(zi-z2) +
dz 2
zl
o dzl
j
1
dz3
'\
dZ 2
j
(4.41)
dz4 M4 M3*M2Me 2 iw(ZlZ2+Z34)
1Z3
1/ A
dZ 3
Z5
dz5
dZ 4
M 6 M5*M 4M3*M2 M1 e 2iw(zl-z2+z3-z4+z5-z6)
T=
1-
dzl
dz2
dz3
dz 2
"A
dz
-
(4.42)
A
z2+ z3-
dz 6
dz 5
- z6 ) +
z4+Z5
dzMe 2iw(z)±
r+
jridzl
+j dziy dZ2y
-|
+
Z4
dz 4
3
M6*M5 M4*M3 M2*Mle- 2 iw(zl-
R=
dzo|
dzJO
dz2 |
d
6
dz4 M4M 3 M2*Mle-2iw(zl-z2+z3-z4)+
rz2
Ai
dzl
-
dz
dz2M*Me- 2iw(zl -z2) +
dz1
+
+
(4.43)
A
2
dZ3 M3*M
2M e i(z z2+z3)+
dz3
Z 2 dz3
|
|
4'dz
dz
94
4
dz5M *M4M M2M e2iw(zl- z2+Z3-z4+z5)
5
5
4
3
R
dzlMle - 2iw(z) +
=- j
dzj
+
dzl
j
(4.44)
dZ2
d
2iw(zZ2+z3) +
ddZ3M3M2Mle
dz2 j
dz3
j
dz4 j4
dz 5M 5 M4M3M2Mle- 2i(zl-z2+z
3- 4+
5)
+...
where Mi = M(zi).The analogous quantities for the antiparticles are obtained replacing M -- M* in all the former formulas.
We also need to have the density matrices pzo and p+.We
can choose these
densities as describing thermal equilibrium densities in eigenstates of the unbroken
phase:
Pzo= Diag(ns(E, v), n,+(E, v), n_ (E, v))
(4.45)
where n(E, v) is the Fermi-dirac distribution, boosted in the wall frame:
n =
and pzo+ = Pzo( -
-v).
e
1(E-vwktr)
TC
(4.46)
+1
The motivation of this is that particles are produced
in interaction eigenstates which differ from mass eigenstates by a unitary rotation;
ignoring this, amounts at ignoring small corrections of order (M(z)/Tc) 2 . The choice
of thermal distribution is particularly good in the small velocity v, regime, in which
we have restricted, where the non thermal contribution is of order v,, and it induces
corrections of order v in the final baryon density [32].
Finally, we have to consider the charges which can play a role in generating the
baryon number. When choosing such charges, one has to consider that the most important charges are those which are approximately conserved in the unbroken phase,
as these are the ones which can efficiently diffuse in the unbroken phase, and induce
a large generation of baryon number. Keeping this in mind, it is easy to see that
the only relevant charge in our model is the "higgs number" charge, which in the
same basis in which we expressed the mass matrix, for the new CP violating sector
95
particles, is given by:
(4.47)
Qh = Diag (O,0, 1, -1)
The name "higgs number" just comes from the fact that the fields I± have the
same quantum numbers as higgsinos in the MSSM. Now, we can substitute in the
formulas for J+ and J_. In order to keep some analytical expression, we decide to do a
derivativeexpansionM(z) = M(zo +M'(zo)(- Zo)+ (r/w)2 and v = + O(r/w)2 .
This expansion is justified in the parameter range r < w, which is expected to be
fulfilled [34].
approximately
Unlike in the case of the analogous calculation for the MSSM [32]where the leading
effect occurs at fourth order in mass insertion, here the leading contribution occurs
at sixth order. We will explain later the reason of this. For the moment, we get:
(4.48)
J+= (1,O,O,v) x (
fz1
fA
Z3
A
+4 A dzl
l
dz2
dZ3 d3 z
sin(2w(zi-4
dz 1
j
z 32+
dz2
j
r
fZ5
dz4
ddz6
Z4 + Z5 - z 6))Im (pZoQhM
6M5*M
4M3*M
2 Mfl)+
dz3
j
dz4
j
f
dz5 J
dz6
sin(2w(zi- 2 - z 3 + z4 - Z5 + z 6))Im (poQhMlM2*M
+
6M5*M
4 M3*)
A
A
sin(2w(z
-
z2 + Z3
A
- z4
A
Z3
Z5
d
+ z5 -z 6))Im (Zo+aQhMM6 M5*M
4MM 2))
96
J_ = (1,0,0, -v) x (
A
A
dZ2odz
dzl J
-4/
(4.49)
A
Z2
dz4
3 j
A
Z4
dz5
dz6
sin(2w(zl - z2 + z3 - z 4 + z5 - z6 ))Im (Pzo+aQhM6 M5 M4 M3M2M) +
+4 j
dzl /
sin(2w(zl r
-4
j
z2
dz3
dz4
dZ3
dz6
dz 5 j
2 - z 3 + z4 - z 5 + z6 ))Im (pZo+AQhMlM2*M6 M5M4M) +
rA
dzl j
A
Z2
dz2 j
dz 3 j
A
Z4
dz4
dz 5 J
dz6
sin(2w(z - Z2 + z3 - z 4 + Z5 - z 6))Im (pQhMlM
6 M5M 4M3M2 ))
where the z dependence in each mass matrix Mi is to be understood at linearized
level.
We can substitute the results in eq.(4.40), to get an expression for the higgs
charge source. In reality, if we take the relaxation time large enough, and if we keep
performing a derivative expansion, only the first term in eq.(4.40) is relevant. At first
order in the wall velocity, we get:
yQ
^- WVW
T7J
7
(Z
where
J
Jt
(4.50)
x
3 k f(wiA)
W+d
3w
7
eEi/T
(1 + eE/T,)2
Ei
/
is a CP violating invariant:
1
2
2
2 - g'2 ) sin(9)
, = -ggkk'(g
4
97
(4.51)
and fi(wA) is:
f(wA)
= v(zO) (z) ((S'(Zo)(9 - 24W2A 2 - 14w4 A4)v(zo) - 9v'(zo)S(zo) (4.52)
24Tj
-2w 2 A(11v'(zo)A(3 + w2A2) + 3(5 + 2w2A 2)v(zo))S(Zo) +
6cos(6wA)(-S'(zo)v(zo) + v'(zo)S(zo)) + 6cos(4wA)(2S'(zo)(-1 + 3w2zA2)v(Zo)
+2v'(zo)S(zo)+ w2 /(-3v'(zo)/A + v(zo))S(zo))+
3 cos(2wA)(3S'(zo)(1
+ 6W 2 A2 )V(ZO)
-3v'(zo)S(Zo) + 2w2 A\(3v'(zo) A + 4V(zo))S(zo))
+2w(2S'(zo)A(-3 + 14W2A 2)v(zo)+ (5v'(zo)A(3+ 7w2A 2) +
3(1 + 7w2 A 2 )v(Zo))S(zo)) sin(2wA)3w(16S'(zo)Av(zo)
+(-13v'(zo)L + v(zo))S(zo)) sin(4wA) + 6wA)
(-S'(zo)v(zo) + v'(zo)S(zo))sin(6wA))))
f,(wA)
(zO)3 S (ZO)(S'(Zo)(15+ 12w2A2 - 28w4
4 )v(
(453o)
-15v'(zo)S(Zo)- 4w2A(v'(zo)A(48 + 11w2A2 ) +
3(5 + 2w2A 2)v(zo))S(Zo) + 12cos(6wA)(S'(zo)v(zo)
-v'(zo)S(zo)) + 3 cos(4wA)(S'(zo)
(1 - 12w2 A 2 )V(Zo) - v'(zo)S(zo) + 4w 2 A(6v'(zo)A + v(zo))S(zoO) +
6cos(2wA(S'(zo)(-5 + 14W2A 2)v(zo) + 5v'(zo)S(zo)+ 2w2A2(5v'(zo)A+
4v(zo))S(zo)) + 12w(S'(zo)A(-7 + 6w2 /X2)(Zo) + ((zo) +
A(5v'(zo)(2 + 3w2LA2 ) + 7W2Av(zo))S(zo))sin(2wA)
-6w(-9S'(zo)Av(zo) + (12v'(zo)A+ v(zo))S(zo)) sin(4wA) + 12wA(S'(zo)v(zo)
-v'(zo)S(zo)) sin(6wA)))
where zo here is a typical point in the middle of the wall. We can estimate the
coherence time as rT
(gT)
-1 ,
and so we can take, roughly, the typical interval
which is valid also in the MSSM [32]: 15 < rT
98
< 35, and in the approximation
of making the particles massless, which is still consistent with our approximations,
we can integrate numerically the integral, and fit it linearly in r (which is a good
approximation in the range of interest), to obtain:
yQ(X, t) _ 24vw-y (gg'kk'(g 2 - g, 2 ) sin(O))
x
(4.54)
(1 + (TTc- 25)) V3S (S)
TTC
T3
Even though the reparametrization
invariant CP violating phase 0 requires only 4
Yukawa couplings to be present in the theory, it is somewhat puzzling that the leading
effect appears at sixth order in the couplings. The reason is as follows. In the case
g = g', there is an approximate Z2 symmetry which exchanges the Xi fields. This
is only an approximate symmetry, because A± differ by their hypercharge. However,
the hypercharge is considered a subleading effect here, and never comes into play
in our computation, and so we do not see the breaking of this symmetry. The Z 2
symmetry has, in the basis we have been using up to now, the matrix form:
Z2=
0
0 1
We clearly see that:
[M(z), Z21 = 0, [p(Z), Z 2] = 0,
{Q, Z 2 } =
0
(4.55)
which then implies that:
Tr (pzoTQTt) = 0
(4.56)
and similar for the others terms in J+ and J_. So, in the limit in which g = g', the
baryon asymmetry should vanish. Then, since each particle needs to have an even
number of mass insertions to be transmitted or reflected, and since the CP violation
requires at least 4 mass insertion, we finally obtain that the coupling dependence in
the CP violating source must be of the form present in eq. (4.54). From this discussion,
99
it is clear that in the case g
=
g' the baryon production will be heavily suppressed,
and, from what we will see later, it will be clear that the couplings in this case will
have to be so large to necessarily hit a Landau pole at very low energies, making the
model badly defined. We shall neglect this degeneracy of the couplings for the next
of the chapter.
Now, we are ready to begin the second part of the computation.
4.3.2
Diffusion Equations
Here, we begin the second part of the calculation, still following [32]. We turn to
analyze the system at a larger scale, and approximate it to a fluid. We then study
the evolution of the CP violating charges due to the presence of the sources and of
the diffusion effects in the plasma. To this purpose, we shall write a set of coupled
differential equations which include the effects of diffusion, particle number changing
reactions, and CP violating source terms, and we shall solve them to find the various
densities. We shall be interested in the evolution of particles which carry some charges
which are approximately conserved in the unbroken phase. Near thermal equilibrium,
which is a good approximation for small velocities, we can approximate the number
density as:
ni = kiiT,2/6
(4.57)
where pi is the chemical potential, and ki is a statistical factor which is equal to 2 for
each bosonic degree of freedom, and 1 for each fermionic.
The system of differential equations simplifies a lot if we neglect all couplings
except the gauge couplings, the top quark Yukawa coupling, and the Yukawa couplings
in the new CP violating sector. From the beginning, we take the interactions mediated
from these last ones to be fast with respect to the typical timescale of the fluid. We
include the effect of strong sphaleron, but neglect the one of the weak sphalerons until
almost the end of the computation. This allows us to forget about leptons. We need
only to keep track of the following populations: the top left doublet: Q = (tL + bL),
the right top: T = tR, the Higgs particle plus our new fields I±:H = ho + + + -I
100
Strong sphalerons will be basically the only process to generate the right bottom
quarks B = bR, and the quarks of the first two generations Q(1,2)L,UR, CR, SR, DR.
This implies that all these abundance can be expressed in terms of the one of B:
Q1L= Q2L= -2UR = -2DR
= - 2 SR
= -2CR = -2B = 2(Q + T)
(4.58)
The rate of top Yukawa interaction, Higgs violating process, and axial top number
violation are indicated as ry, rh, rm, respectively. We take all the quarks to have
the same diffusion equation, and the same for the higgs and the s. The charge
abundances are then described by the following set of differential equations:
Q=
DqV2Q - ry(Q/kQ - H/kH - T/kT) - rm(Q/kQ - T/kT)
(4.59)
-6r,,(2Q/kQ - T/kT + 9(Q+ T)/kB)
=
T
DqV2T - ry(-Q/kQ + H/kH + T/kT)
-rm(-Q/kQ +T/kT)+ 3r.,(2Q/kQ- T/kT+ 9(Q+T)/kB)
H=
DhV2H - ry(-Q/kQ + T/kT + H/kH) - rhH/kh + YQ
We can restrict ourselves to the vicinity of the wall, so that we can neglect the
curvature of the surface, and assume we can express everything in a variable z
IrF+ vt.
=
The resulting equations of motions become:
vwQ'
DqQ" - F(Q/kQ - H/kH - T/kT) - rm(Q/kQ - T/kT)
(4.60)
-6r,,(2Q/kQ - T/kT + 9(Q+ T)/kB)
vWT'=
DqT"- r(-Q/kQ + H/kH + T/kT)
-rm(-Q/kQ + T/kT) + 3r,,(2Q/kQ- T/kT + 9(Q+ T)/kB)
VWH'=
DhH" - ry(-Q/kQ + T/kT + H/kH) - rhH/kh + YQ
We now assume that ry and r,, are very fast, and we developthe result at O(1/ry, 1/r,,).
This allows to algebraically express Q and T in terms of H, to get the following re-
101
lationships:
Q=H(
kQ(9kT-kB)
'=k
kH(kB + 9kQ + 9kT))
)(4.61)
(4.61)
and,
find)
substituting
back,we
equation for H:
(4.62)differential
and, substituting back, we find the following effective differential equation for H:
vH' = DH"- rH +5
(4.63)
where the effective couplings are given by:
D = Dq(9kQkT - 2kQkB - 2kBkT) + DhkH(9kQ + 9kT + kB)
9kqkT - 2kQkB - 2kBkT + kH(9kQ + 9kT + kB)
9kkT-
kH(9 kQ + 9kT + kB)
,Q99kQkT
- 2kQkB
-2kBkT
+kH(9kQ
+9kT
+kB
r = (rm +rh)
91cc
(9kQ + 9kT + kB)
9kQkT - 2kQkB - 2kBkT + kH(9kQ+ 9kT + kB)
(4.64)
(4.65)
(4.66)
We can estimate the relaxation rates for the higgs number and the axial quark
number as [32]:
(m
+ Fh)
4M1 ,(T,
z)
21g2T
A
(4.67)
where At is the top Yukawa coupling, and Mw is the W boson mass, and, in order to
keep analytical control, we approximate the source term and the relaxation term as
step functions:
= 5', w >
> O
(4.68)
y = O, otherwise
and
r=r,
F=o,
102
z0>o
<0O
(4.69)
For the source term j, we can take the avaraged value of expression (4.54). However,
due to our lack of knowledge of the details of the profiles of the fields during the phase
transition, we can just approximate that expression with:
yQ =
24vwyw (gg'kk'(g
2
2 ) sin(O)) x
- g9'
(4.70)
- 25)) v4S2
rT
T3 W
(T,
(1+
where we have taken S' . S/W and v'
v/W, with W the wall width, and we have
-
assumed, as expected, that no cancellation is occurring. Here S and v are taken to
be of the order they are today. In the approximation that D is constant, and with
the boundary conditions given by H(±oo) = 0, we have an analytical solution in the
unbroken phase [32]:
H = AeZVw/D
(4.71)
where, in the limit that DI < vW, which is in general applicable,
A-
(4.72)
(1 - e-2WV))
Note that diffusion of the higgs field, and so of the other charges, in the unbroken
phase, occurs for a distance of order z
D/vw.
We now turn on the weak sphaleron rate, which is the responsible for the baryon
generation. The baryon density follows the following equation of motion:
VwPB = Dqp - E(-f)3rw,nL()
(4.73)
where we have assumed that the weak sphaleron operates only in the unbroken phase,
and where nL is the total number density of left fermions.
The solution to this
equation is given by, at first order in vw:
PB = -
W
w
dznL()
-oo
103
(4.74)
Now, in the approximation in which all the particles in our theory are light, we have:
kQ = 6, kT = 3, kB = 3, kH = 8
(4.75)
and in the limit as rF --, oo, the resulting baryon abundance is zero [32]. This
means that we have to go to the next order in the strong sphaleron rate expansion.
Note also that, with this particle content, using the SM quark and higgs diffusion
equation Dq
6/T, Dh
'.
96/Tc and DFr/lv > 2/v,
110/T [35],we have D
so that the assumption that the Yukawa interaction is fast is self-consistent, since
ry
(27/2)At2asTc [32]. Finally, we take:
rF7 = 6k'3-a 4T
rws = 6kaoTc,
(4.76)
where k' is an order one parameter and k - 20 [25]. To go to next order in the
expansion in large F,,, we write:
=
Q
- kB) )
Q=H( k( +9kT
(k
Substituting
in (4.60), we get:
H
++9k
(2kQ
B+
))
+
kQ
Q
5QI=F-v '
JQ
= (DH
Using nL
=
Q+
H
(4.78)
(4.78)
(4.79)
(1/)
3kH(9kQ + 9kt + kB)2
Q1L + Q2L = 5Q + 4T
nL =
(4.77)
(5kQ+4kT) 6 Q,
-
we get:
(D H -vwH)
76 = -
(479)
(4.80)
Substituting the solution for the Higgs field, we finally get:
PB
__
(112
s
r
(A1s
104
)(1
Dq
(4.81)
where s = (27r2 g,/45)Tg - 55T3 . Substituting our parameters, we get:
PB
s
5x
10-
5
( k,p/20
k
(gg(g 2 - g'2)kk' sin(O))
1(T - 25))
(1 + (8T/25)
(T,/25)
(T
(4.82)
(S
It is worth to make a couple of small comments on the parametric dependence of
this expression. The dependence of the coupling terms, and therefore on the vevs
of h and S, was explained in the former section. The wall width W has simplified
away because the exponential in eq.(4.72) is small and can be Taylor expanded. The
presence of
s,, in the denominator is due to the particular particle content of this
model, for which the leading term in the expansion in 1/r,, is zero. The factor 10- 5 is
mainly due to the factor of - 10-2 from the entropy density, and the ratio 20 a 5 /4,
times some other factors coming from the diffusion terms. For typical values of the
wall velocity, we can take approximately the SM range: v
- 0.05 - 0.3, while the
mean free time is -T- 20/To - 30/Tc. With these values, the produced baryon number
ranges in the regime:
PB
(2 X 10
-
-
3 x 10 - 6) (gg(g
(
2
)
_ g' 2 )kk' sin(O))
(4.83)
T1 T, T,)
(4.83)
This is the number which has to be equal to the baryon density at BBN:
= 9
(PB)
S
BBN
X 10"
(4.84)
For the moment, let us try do draw some preliminary conclusions on what this does
imply on the parameters of the model. Clearly, there are still too many parameters
which could be varied, so, as a beginning, we can set all the Yukawa couplings in
the new CP violating sector to be roughly equal to each other k -
'- g
g', and
v/T - 1, we then get:
PB , (2 x 10 S
7
- 4 x 10- 6) (g6 sin(O))
$
105
-
1010
(4.85)
We then get the following constraint:
10- 4
g 6 (-)sin(O)
(4.86)
It is natural, in this theory, to take the CP violating phase to be of order 1. In this
case:
g
-g'
k
k'
0.21()
(4.87)
1/
From this we see that the assumption of considering the interactions mediated by
these Yukawa couplings to be fast is justified for a large fraction of the parameter
space. These values become lower bounds for the couplings if we allow for the CP
violating phase to be smaller than order 1 (see eq.(4.86)).
4.4
Electric Dipole Moment
The same CP violating phase which is responsible for baryogenesis, induces an electric
dipole moment through the 2-loop diagram shown in fig.4-1.
f'
f
Figure 4-1: 2 loops diagram contributing to fermion EDM, where Xi are the neutral mass
eigenstates, and w+ is the charged one.
The situation here is much different than in the MSSM, where a CP violating phase
in general introduces EDM at one loop level, the constraints on which generically force
very small CP violating phases in the MSSM. It is instead much more similar to the
case of split supersymmetry [44, 45, 47], where all the one loop diagrams contributing
to the EDM are decoupled. Here the leading diagram is at two loops level, and, as we
will soon see, it will induce electron EDM naturally just a little beyond the present
106
constraints, and on the edge of detection by future experiments.
The induced EDM is (see [38]):
.
e
±
2 mf
872sM 2
(r, r, rf,)
1'dy o[1
±mx
o dzo' dy
-IjZy
[ (R-
rf)
Im (LOR*)9 (rir,
M(4.8)
dy
(4.88)
,
3(z
z/2)
(zyz(y+
+R)
+K i)
3Ki)R + 2(Ki + R)y
4R(Ki - R) 2
Ki(Ki - 2y)
Ki
2(Ki- R) 3
R
(4.89)
with
w
R = y+ ( - y)rf,, Ki = 1-
M , rI
OR = N3*i,O = -N 4 iC R
where CR = e - i° and NTMNN
2
°
M2
f
. (4.90)
(4.91)
= diag{mX
,MX,m,m
1
3} with real and positive
diagonal elements. The plus(minus) sign on the right-hand side of eq. (6.2) corresponds
to the fermion f with weak isospin +(-)1/2.
and f' is its electroweak partner. We
mean by w the charged mass eigenstate, and with Xi, i = 1, 2, 3, the neutral mass
eigenstates in order of increasing mass.
Diagonalizing the 3 x 3 mass matrix numerically, we see that generically the model
predicts EDM very close to detection. The induced EDM for some generic parameters
are shown in fig.4-2.
It is clear then that improvements of the determination of the EDM are going
to explore the most interesting region of the parameter space. Ongoing and next
generation experiments plan to improve the EDM sensitivity by several orders of
magnitude within a few years. For example, DeMille and his Yale group [39] will use
the molecule PbO to improve the sensitivity of the electron EDM to 10-29 e cm within
three years, and possibly to 10- 3 1 e cm within five years. Lamoreaux and his Los
Alamos group [40] developed a solid state technique that can improve the sensitivity
107
de/(e cm)
l
.
I
1. x 10-27
1. x 10- 28
1. x 10- 29
.
-
1. x 10
- 30
100
g=9'=O.1
..... '
'
______________________________________
300
500 700 1000 1500 2000
150 200
Figure 4-2: The predicted EDM in this model. We plot the induced electron EDM as
a function of k'S. The solid line represents the induced EDM with g = g' = 1/2 and
kS = 100 GeV, while the dashed line represents g = g' = 1/10 and kS = 100 GeV, and we
take maximal CP violating phase. The horizontal line represents the present electron EDM
constraint de < 1.7 x 10-27 e cm at 95% CL [37].
to the electron EDM by 104to reach 10-31 e cm. By operating at a lower temperature
it is feasible to eventually reach a sensitivity of 10- 3 5 e cm, an improvement of eight
orders of magnitude over the present sensitivity. The time scale for these is uncertain,
as it is tied to funding prospects. Semertzidis and his Brookhaven group [41] plan to
trap muons in storage rings and increase the sensitivity of their EDM measurement
by five orders of magnitude. A new measurement has been presented by the Sussex
group [42]. A number of other experiments aim for an improvement in sensitivity by
one or two orders of magnitude, and involve nuclear EDMs.
In order to understand the current and future constraints on the model, we can
study what is the induced electron EDM, once we have satisfied the constraint from
baryogenesis in eq.(4.83):
99(92 _
g'2 )kk'
v3 S2
,, 10- 4
(4.92)
c
The way we proceed is as follows. First, we fix g' = g/x/.
This is a good represen-
tative of the possible ratios between g and g', as it is quite far from the region where
108
the approximate symmetry in the case g = g' suppresses baryogenesis, and g' is not
too small to suppress baryogenesis on its own. Later on we shall relax this condition.
Having done this, we invert eq.(4.92), to get an expression for g in terms of the other
parameters of the model. Now, the couplings g, g' are expressed in terms of T, and
in terms of kS and k'S which, as it will be useful, in the case of small mixing, can
be thought of as respectively the singlet and the doublet mass. We shall impose the
constraint on the couplings g, g', k, k' to be less than - 1, in order for these Yukawa
couplings not to hit a Landau pole before the unification scale
1015GeV.
We decide to restrict our analysis to the case in which the possible ratios between
the marginal couplings of the same sector of the theory are smaller than 2 orders of
magnitude. Here, by same sector of the theory we mean either the CP violating sector,
or the sector of the scalar potential and the strong gauge group. The justification
of this relies in the fact that we wish to explore the most natural region of the
parameter space. For this reason, we expect that there is no large hierarchy between
the marginal couplings of the same sector. We think that 2 order of magnitudes is
a threshold large enough to delimitate this natural region. However, since marginal
couplings are radiatively relatively stable, and large hierarchies among them does
not give rise to fine tuning issues, in principle, large hierarchies among the marginal
couplings are acceptable. We think that, however, such a hierarchy would require the
addition of further structure to the model to justify its presence, and we decide to
restrict to the simplest realization of the model. As a consequence of this, the most
important restrictions we apply are: 10- 2 < k/k' < 102, and 10-2 < kslA < 102. In
particular, using eq.(4.33) and eq.(4.34), this implies
T < S
5 T,, which, using
the limit k, k' < 1, implies that the largest of kS and k'S must be smaller than 5 T.
The upper bound on the CP violating sector particles tells us that there is a small
part of the parameter space which we are going to explore, in which the particles
responsible for the production of baryons are non relativistic. In that case, we can
approximately extend the result found in eq.(4.92), with the purpose of having order
of magnitude estimates, in the following way. From eq.(4.50), it is clear that baryon
production is Boltzmann suppressed if the particles are non relativistic. In that case,
109
the CP violating charge will be generated by the scattering of these particle in the
region where the induced mass on the particles is of the order of the critical temperature, as this is the condition of maximum CP violating interaction compatible
with not being Boltzmann suppressed. Having observed this, it is easy to approximately extend the result of eq.(4.92) to the non relativistic case, by taking the vevs
of the fields at the a value such that the induced mass is of the order of the critical
temperature.
In fig.3, we show the induced EDM as a function of kS, for several values of
the critical temperature T, for k'S = 100, on top, and k'S = 500, at the bottom,
with the couplings g, g' chosen as explained above, in order to fulfil the baryogenesis
requirement. Notice that 500 GeV is roughly the limit that LHC will put on SU(2)
doublets. We choose the maximum CP violating phase. The horizontal lines represent
the present constraint on EDM de < 1.7 x 10-2 7 e cm at 95% CL [37], and the future
expected one [39, 40, 41] of order 10-3 1 e cm. A few features are worth to be noted.
We see that, for fixed T, the EDM decreases as we decrease kS, for kS light enough.
This is due the fact that, reducing kS, we reduce both the EDM and the produced
quantity of baryons. However, the loss in the production of baryons is compensated
with a much smaller, compared to the decrease in kS, increase in the couplings g, g',
so that, the baryon abundance can remain constant, while the EDM decreases. For
large
S, the EDM decreases both because the mass of the particles in the loops
becomes heavier and heavier, and also because the mixing becomes more and more
suppressed. The maximum is located at the point where kS
k'S. In that case,
in fact, the mass matrix has a diagonal piece roughly proportional to the identity,
so, even though the diagonal elements are much larger than the off diagonal ones,
mixing is much enhanced, and so is the EDM, which is proportional to the mixing.
The lower and upper limit on the value of kS, as well as the minimum temperature,
are dictated by the restrictions k'S < 5 T, kS < 5 T, and k/k' > 10- 2 . Now, as
we decrease the critical temperature, the resulting EDM tends to decrease. In fact,
decreasing the temperature much enhances the baryon production.
This allows to
decrease the couplings g, g', which explains why the induced EDM decreases. We
110
verify that increasing the hierarchy between g and g' does not change the results
a lot, as the decrease in baryons production requires to make the other couplings
large. Increasing the value of k'S up to the maximum allowed value of 1.1 TeV (since
T.m
a
-
v) does not produce any relevant change in the result, because this raises the
minimum temperature, so that baryogenesis requires larger couplings, which forces
the EDM not to decrease relevantly. It is only once the CP violating phase is lowered
to less than 10-2 that a small experimentally unreachable region is opened up around
k'S = 500 GeV and kS < 1 GeV. The same region is opened also enlarging the
allowed hierarchy between the couplings to above 5 x 102. However, clearly, the
presence at the same time of a large hierarchy and of a very small CP phase requires
some additional structure on the model to explain the reason for their presence.
The main conclusion we can draw from combining the analysis on baryogenesis
and EDM is that, at present, the most natural region of the parameter space is
perfectly allowed, however, improvements in the determination of the EDM, are going
to explore the entire viable region of the parameter space, so that absence of signal,
would result in effectively ruling out the model, at least if no further structure is
added.
4.5
Comments on Dark Matter and Gauge Coupling Unification
The proposed model finds its main motivation in stabilizing the weak scale through
the requirement of attaining baryons in our universe. However, further than this, it is
clear that the model provides two other interesting phenomenological aspects: gauge
coupling unification and dark matter. Here, we just briefly introduce these aspects,
and we postpone a more detailed discussion to future work.
Thanks to the two doublets we have inserted in our model, which have the same
quantum numbers as higgsinos in the MSSM, gauge coupling unification works much
better than in the standard model. As it was shown in [11], gauge coupling unification
111
with the standard model plus higgsinos works roughly as well as the MSSM at two
loops level, with the only possible problem being the fact that the unification scale
is a bit low at around
1015 GeV. The problem from proton decay can however
be avoided with some particular model at the GUT scale [11, 43]. We expect that
2-loop gauge coupling unification works quite well also in this model, with only small
corrections coming from the presence of the singlet scalar S.
Concerning the Dark Matter relic abundance, the lightest of the newly introduced
particles is stable, and so it provides a natural candidate for Dark Matter. Estimating
the relic abundance is quite complex, as it depends on the composition of the particle,
and on its annihilation and coannihilations rate. However, if our newly introduced
Yukawa couplings are not very close to their upper limit (of order one), and if the
lightest particle is mostly composed of the two doublets, then its relic abundance is
very similar to the one of pure higgsino dark matter in split supersymmetry [44, 45, 46,
47, 48, 49], requiring the doublets to be around the TeV scale. In the case of singlet
dark matter, we expect the singlet to give the right abundance for a much lighter
mass, as it is naturally much less interacting. However, estimates become much more
difficult, and we postpone the precise determination of the relic abundance to future
work.
4.6
Conclusions
In this study, we have addressed the solution to the electroweak hierarchy problem in
the context of the landscape, following a recent model proposed in [11].
We have shown that it is possible to connect the electroweak scale to a hierarchically small scale at which a gauge group becomes strong by dimensional transmutation. The assumption is that we are in a "friendly neighborhood" of the landscape in
which only the relevant parameters of the low energy theory are effectively scanned.
We then realize the model in such a way that there is a fragile, though necessary, feature of the universe which needs to be realized in our universe in order to sustain any
sort of life. As a natural continuation of Weinberg's "structure principle", the fragile
112
feature is the presence of baryons in the universe, which are a necessary ingredient for
the formation of clumped structures. We assume that, in the friendly neighborhood
of the landscape in which we should be, baryogenesis is possible only through the
mechanism of electroweak baryogenesis. Then, in order to produce a first order phase
transition strong enough so that sphalerons do not wash out the produced baryon
density, we develop a new mechanism to implement the electroweak phase transition.
We introduce a new gauge sector which becomes strong at an exponentially small
scale through dimensional transmutation.
We couple this new sector to a singlet S
which is then coupled to the higgs field. The electroweak phase transition occurs
as the new gauge sector becomes strong, and produces a chiral condensation. This
triggers a phase transition for the singlet S, which then triggers the phase transition
for the higgs fields. In order to preserve baryon number, we need the phase transition
to be strong enough, and this is true only if the higgs mass is comparable to the QCD
scale of the strong sector. This solves the hierarchy problem. In order to provide the
necessary CP violation, we introduce 2 SU(2) doublets
t with hypercharge ±1/2,
and a gauge singlet s.
When we require the model to describe our world, the model leads to falsifiable
predictions.
The main result of the study in this chapter is the computation of the produced
baryon number, for which we obtain:
PB_
(2 x 10
3 x100 6) (9gg(g2
g'2 )kk' sin(0))
()
(S
(4.93)
The requirement that this baryon abundance should cope with the observed one leads
to a lower bound on a combination of the product of the CP violating phase, the new
couplings, the S vev, and the critical temperature Tc.
We infer that Gauge Coupling Unification, and the right amount of Dark Matter
relic abundance are easily achieved in this model.
We study in detail the induced electron Electric Dipole Moment (EDM), and we
find that, at present, the most natural region of the parameter space of the theory is
113
allowed. However, soon in the future improvement in the EDM experiments will be
sensitive the entire viable region of the parameter space of the model, so that absence
of a signal would result in practically ruling out the model.
4.7
Appendix: Baryogenesis in the Large Velocity
Approximation
The method we have used in the main part of this work is based on some approximations that fail in limit of very fast wall speed: v,
1. Even though this regime
seems to be disfavored by actual computations of the wall speed[31], it is worth to
try to estimate the result even in this case. In this appendix we are going to do
an approximate computation in the large wall velocity approximation.
In this new
regime, calculations become much more complicated, as the assumption of local thermodynamical equilibrium begins to fail, and the baryon number tends to be produced
in the region of the wall, in the so called local baryogenesis scenario.
We follow the treatment of [51]. In this approach, the phase transition is treated
from an effective field theory point of view. A good way of looking at the high
temperature phase of the unbroken phase is imagining it as discretized in a lot of
cells, whose side is given by the typical size of the weak sphaleron barrier crossing
configuration of the gauge field:
(awT)
- 1.
Concentrating on each cell, we
have that the thermal energy in a cell is in general much larger than the energy
necessary to create a gauge field oscillation capable of crossing the barrier, and this
means that most of the energy is in oscillations with smaller wavelength than (. So,
these configurations cross the barrier at energy far above the one of the sphaleron
configuration, and so their rate has nothing to do with the sphaleron rate, and this
explain why their rate per unit volume is of order 4 (In the case m2 < 0, electroweak
symmetry breaking has already occurred outside the bubble, but this discussion still
roughly applies). We can parameterize the configuration in one cell with one variable
T,
which depends only on time. Obviously, this is a very rough approximation, but
114
this is at least a beginning. We try to describe the dynamics of the configuration near
crossing the barrier, at a maximum of the energy, that we fixe to be at r = 0. Near
this point, we can write the following Lagrangian, which is very similar to the one in
[51]:
L(T, *)= cl
2
+ C2
2C
2
+ C3b
(4.94)
2(3
where ci are dimensionless parameters depending on the different possible barrier
crossing trajectories, while b will be determined shortly. Since the point r = 0 is
a maximum of the energy, no odd powers of r can appear. The appearance of the
odd power in i can be understood because, at one loop level, the CP violating mass
matrix of the particles in the CP violating sector induces an operator which contains
the term TrF,,F/".
This can be seen in the following way, in an argument similar
to the one shown in [47]. We can imagine to do a chiral rotation on these field of
amplitude equal to the CP violating phase. Because of the anomaly, this will induce
the operator:
(4.95)
= 16w
9222TrFFWv
where g2 is the SU(2) weak coupling, and F,, is the SU(2) field strength, and where
v
2 sin())
Arg(Det(M)) = Arcsin (kk'S
+ (gg
kkIS2sin()2 + (gg'v2 + kkS2Cos())2
(4.96)
Promoting the vevs of h and S to the actual fields, we find the operator we were look-
ing for. Now, the term TrF,,vFi contains a term proportional to the time derivative
of the Chern Simons number, and so the operator
action (4.94) with a term proportional to b,
must contribute to the effective
with b =
g
16r2 ,
explaining the reason
for the presence of the term in i
The equation of motion for r is:
=
C2_
C1
2
27--
115
C3 92g4
c 1 16wr2
(4.97)
If the wall is fast enough,we can solve it in the impulse approximation, to get:
nAi
=
c
3
cl 16w2
g
(4.98)
This kick to A-? makes the distribution of velocities of barrier crossing configurations
asymmetric, leading to a production of baryons with respect to antibaryons.
Now, this kick will be very inefficient in changing the distribution of baryons,
unless the kick is larger than the typical speed o a generic configuration would have
if it crossed the barrier in the absence of the wall. So, we require: A
> '0. The
fraction of configurations which satisfy this requirement is proportional to Ai, but is
very difficult to estimate. Following [51], we just say that it is equal to f A, where
f is our "ignorance" coefficient. So, we finally get:
nB
f
-( 3
(4.99)
where we have reabsorbed the constants ci into f. We finally get:
PB _afW3
ss
f a4455
92
6 22 AO
167r
(4.100)
There are a lot of heavy approximations which suggest that this estimate is very rough
[51]: from the value of the coefficient ci, to the approximation of restricting to one
degree of freedom, to the impulse approximation in solving the differential equation,
and to the estimate of the fraction of configurations influenced by the kick. All these
suggests that we should take eq.(4.100) with f
baryon production, as the authors of [51] suggest.
116
1, as at most an upper limit on the
de/(e cm)
1. x 10-27
_
I
7
-
1. x 10 28
1. x 10- 29
1. x 1030
1. x10 - 31
.
T=25 GeV . -5
10
1
S
50
100
500 1000
de/(e cm)
1. x 10- 27
i
k'S=500 GeV
1. x 10 28
1.xl 0-29
T=200 GeV
1.x 1 0-30
1.x 10- 31
I,
c
K
10
20
50
100
200
500
1000
Figure 4-3: Top, the predicted EDM given the constraint from Baryogenesis satisfied,
as a function of kS for kS = 100 GeV, g' = gl/,
maximum CP violating phase, and
T, = 200 GeV (solid line), 100 GeV (long dashed), 25 GeV (short dashed). Bottom, the
same for k'S = 500 GeV, and T = 200 GeV (solid line), 150 GeV (long dashed), 110
GeV (short dashed). The horizontal lines represent the present electron EDM constraint
de < 1.7 x 10-2 7 e cm at 95% CL [37], and the expected improvement up to de < 103 1e cm
[39, 40, 41].
117
Bibliography
[1] R. Barbieri and A. Strumia, "What is the limit on the Higgs mass?," Phys. Lett.
B 462 (1999) 144, hep-ph/9905281.
[2] L. Giusti, A. Romanino and A. Strumia, "Natural ranges of supersymmetric signals," Nucl. Phys. B 550 (1999) 3, hep-ph/9811386.
[3] D. N. Spergel et al. [WMAP Collaboration], "First Year Wilkinson Microwave
Anisotropy Probe (WMAP) Observations: Determination of Cosmological Parameters," Astrophys. J. Suppl. 148 (2003) 175, astro-ph/0302209.
[4] S. Perlmutter et al. [Supernova Cosmology Project Collaboration], "Measurements
of Omega and Lambda from 42 High-Redshift Supernovae," Astrophys. J. 517
(1999) 565, astro-ph/9812133.
[5] R. Bousso and J. Polchinski, "Quantization of four-form fluxes and dynamical neutralization of the cosmological constant," JHEP 0006 (2000) 006, hep-th/0004134.
[6] S. B. Giddings, S. Kachru and J. Polchinski, "Hierarchies from fluxes in string
compactifications," Phys. Rev. D 66 (2002) 106006, hep-th/0105097.
[7] G. Curio and A. Krause, "G-fluxes and non-perturbative stabilisation of heterotic
M-theory," Nucl. Phys. B 643 (2002) 131, hep-th/0108220.
[8] A. Maloney, E. Silverstein and A. Strominger, "De Sitter space in noncritical
string theory," hep-th/0205316.
[9] S. Kachru, R. Kallosh, A. Linde and S. P. Trivedi, "De Sitter vacua in string
theory," Phys. Rev. D 68, 046005 (2003), hep-th/0301240.
118
[10] L. Susskind, "The anthropic landscape of string theory," hep-th/0302219.
[11] N. Arkani-Hamed, S. Dimopoulos and S. Kachru, "Predictive landscapes and
new physics at a TeV," hep-th/0501082.
[12] M. R. Douglas, "The statistics of string / M theory vacua," JHEP 0305 (2003)
046, hep-th/0303194.
[13] G. Dvali and A. Vilenkin, "Cosmic attractors and gauge hierarchy," Phys. Rev.
D 70 (2004) 063501, hep-th/0304043.
[14] S. Ashok and M. R. Douglas, "Counting flux vacua," JHEP 0401 (2004) 060,
hep-th/0307049.
[15] F. Denef and M. R. Douglas, "Distributions of flux vacua," JHEP 0405 (2004)
072, hep-th/0404116.
[16] A. Giryavets, S. Kachru and P. K. Tripathy, "On the taxonomy of flux vacua,"
JHEP 0408 (2004) 002, hep-th/0404243.
[17] J. P. Conlon and F. Quevedo, "On the explicit construction and statistics of
Calabi-Yau flux vacua," JHEP 0410 (2004) 039, hep-th/0409215.
[18] 0. DeWolfe, A. Giryavets, S. Kachru and W. Taylor, "Enumerating flux vacua
with enhanced symmetries," JHEP 0502 (2005) 037, hep-th/0411061.
[19] J. Kumar and J. D. Wells, "Landscape cartography: A coarse survey of gauge
group rank and stabilization of the proton," Phys. Rev. D 71 (2005) 026009,
hep-th/0409218.
[20] G. Dvali, "Large hierarchies from attractor vacua," hep-th/0410286.
[21] R. Blumenhagen, F. Gmeiner, G. Honecker, D. Lust and T. Weigand, "The
statistics of supersymmetric D-brane models," Nucl. Phys. B 713 (2005) 83, hep-
th/0411173.
119
[22] F. Denef and M. R. Douglas, "Distributions of nonsupersymmetric flux vacua,"
JHEP 0503 (2005) 061, hep-th/0411183.
[23] S. Weinberg, "Anthropic Bound On The Cosmological Constant," Phys. Rev.
Lett. 59 (1987) 2607.
[24] P. C. Schuster and N. Toro, "Is dark matter heavy because of electroweak symmetry breaking? Revisiting heavy neutrinos," hep-ph/0506079.
[25] D. Bodeker, Phys. Lett. B426 (1998) 351, hep-ph/9801430; P. Arnold and
L.G. Yaffe, hep-ph/9912306;
P. Arnold,
Phys. Rev. D62 (2000) 036003;
G.D. Moore and K. Rummukainen, Phys. Rev. D61 (2000) 105008; G.D. Moore,
Phys. Rev. D62 (2000) 085011.
[26] A. D. Sakharov, "Violation Of CP Invariance, C Asymmetry, And Baryon Asymmetry Of The Universe," Pisma Zh. Eksp. Teor. Fiz. 5 (1967) 32 [JETP Lett. 5
(1967 SOPUA,34,392-393.1991 UFNAA,161,61-64.1991) 24].
[27] A. Riotto, "Theories of baryogenesis," hep-ph/9807454.
[28] A. Riotto and M. Trodden, "Recent progress in baryogenesis," Ann. Rev. Nucl.
Part. Sci. 49 (1999) 35, hep-ph/9901362.
[29] M. B. Gavela, P. Hernandez,
J. Orloff, O. Pene and C. Quimbay,
"Standard
model CP violation and baryon asymmetry. Part 2: Finite temperature," Nucl.
Phys. B 430, 382 (1994), hep-ph/9406289.
[30] K. Rummukainen, M. Tsypin, K. Kajantie, M. Laine and M. E. Shaposhnikov,
"The universality class of the electroweak theory," Nucl. Phys. B 532, 283 (1998),
hep-lat/9805013.
[31] M. Dine, R. G. Leigh, P. Y. Huet, A. D. Linde and D. A. Linde, "Towards
the theory of the electroweak phase transition," Phys. Rev. D 46 (1992) 550,
hep-ph/9203203.
120
[32] P. Huet and A. E. Nelson, "Electroweak baryogenesis in supersymmetric models,"
Phys. Rev. D 53 (1996) 4578, hep-ph/9506477.
[33] A. Riotto, "Towards a Nonequilibrium Quantum Field Theory Approach to Electroweak Baryogenesis," Phys. Rev. D 53 (1996) 5834, hep-ph/9510271; A. Riotto, "Supersymmetric electroweak baryogenesis, nonequilibrium field theory and
quantum Boltzmann equations," Nucl. Phys. B 518 (1998) 339, hep-ph/9712221;
A. Riotto, "The more relaxed supersymmetric electroweak baryogenesis," Phys.
Rev. D 58 (1998) 095009, hep-ph/9803357.
[34] P. Huet and E. Sather, "Electroweak baryogenesis and standard model CP violation," Phys. Rev. D 51, 379 (1995), hep-ph/9404302.
[35] M. Joyce, T. Prokopec and N. Turok, "Nonlocal electroweak baryogenesis. Part
1: Thin wall regime," Phys. Rev. D 53 (1996) 2930, hep-ph/9410281.
[36] X. Zhang and B. L. Young, "Effective Lagrangian approach to electroweak baryogenesis: Higgs mass limit and electric dipole moments of fermion," Phys. Rev. D
49 (1994) 563, hep-ph/9309269;
[37] B. C. Regan, E. D. Commins, C. J. Schmidt and D. DeMille, "New limit on the
electron electric dipole moment," Phys. Rev. Lett. 88 (2002) 071805.
[38] D. Chang, W. F. Chang and W. Y. Keung, "Electric dipole moment in the split
supersymmetry models," Phys. Rev. D 71 (2005) 076006, hep-ph/0503055.
[39] D. Kawall, F. Bay, S. Bickman, Y. Jiang and D. DeMille, "Progress towards
measuring the electric dipole moment of the electron in metastable PbO," AIP
Conf. Proc. 698 (2004) 192.
[40] S. K. Lamoreaux, "Solid state systems for electron electric dipole moment and
other fundamental measurements," nucl-ex/0109014.
[41] Y. K. Semertzidis, "Electric dipole moments of fundamental particles," hepex/0401016.
121
[42] J. J. Hudson, B. E. Sauer, M. R. Tarbutt and E. A. Hinds, "Measurement of
the electron electric dipole moment using YbF molecules," Phys. Rev. Lett. 89
(2002) 023003, hep-ex/0202014.
[43] R. Mahbubani,
L. Senatore, in preparation
[44] N. Arkani-Hamed and S. Dimopoulos, "Supersymmetric unification without
low energy supersymmetry and signatures for fine-tuning at the LHC," hep-
th/0405159.
[45] G. F. Giudice and A. Romanino, "Split supersymmetry," Nucl. Phys. B 699
(2004) 65 [Erratum-ibid.
B 706 (2005) 65], hep-ph/0406088.
[46] A. Pierce, "Dark matter in the finely tuned minimal supersymmetric standard
model," Phys. Rev. D 70 (2004) 075006, hep-ph/0406144.
[47] N. Arkani-Hamed, S. Dimopoulos, G. F. Giudice and A. Romanino, "Aspects of
split supersymmetry," Nucl. Phys. B 709 (2005) 3, hep-ph/0409232.
[48] L. Senatore, "How heavy can the fermions in split SUSY be? A study on gravitino
and extradimensional LSP," Phys. Rev. D 71, 103510 (2005), hep-ph/0412103.
[49] A. Masiero, S. Profumo and P. Ullio, "Neutralino dark matter detection in split
supersymmetry scenarios," Nucl. Phys. B 712 (2005) 86 [arXiv:hep-ph/0412058].
[50] see for example E. W. Kolb and M. S. Turner, The Early Universe, Perseus
Publishing,
1990
[51] A. Lue, K. Rajagopal and M. Trodden, "Semi-analytical approaches to local
electroweak baryogenesis," Phys. Rev. D 56 (1997) 1250, hep-ph/9612282.
122
Chapter 5
Limits on non-Gaussianities from
WMAP data
We develop a method to constrain the level of non-Gaussianity of density perturbations when the 3-point function is of the "equilateral" type. Departures from Gaussianity of this form are produced by single field models such as ghost or DBI inflation
and in general by the presence of higher order derivative operators in the effective
Lagrangian of the inflaton. We show that the induced shape of the 3-point function
can be very well approximated by a factorizable form, making the analysis practical.
We also show that, unless one has a full sky map with uniform noise, in order to saturate the Cramer-Rao bound for the error on the amplitude of the 3-point function,
the estimator must contain a piece that is linear in the data. We apply our technique
to the WMAP data obtaining a constraint on the amplitude fuil
of "equilateral"
non-Gaussianity: -366 < fNqUil.< 238 at 95% C.L. We also apply our technique to
constrain the so-called "local" shape, which is predicted for example by the curvaton
and variable decay width models. We show that the inclusion of the linear piece in
the estimator improves the constraint over those obtained by the WMAP team, to
-27 < fLcal < 121 at 95% C.L.
123
5.1
Introduction
Despite major improvements in the quality of cosmological data over the last few
years, there are very few observables that can characterize the early phases of the Universe, when density perturbations were generated. Deviations from a purely Gaussian
statistics of density perturbations would be a very important constraint on models
of early cosmology. In single field slow-roll inflation, the level of non-Gaussianity
is sharply predicted [1, 2] to be very small, less than 10-6. This is quite far from
the present experimental sensitivity. On the other hand, many models have recently
been proposed with a much higher level of non-Gaussianity, within reach of present
or forthcoming data. For nearly Gaussian fluctuations, the quantity most sensitive to
departures from perfect Gaussianity is the 3-point correlation function. In general,
each model will give a different correlation between the Newtonian potential modes1 :
(~(kl)(k2)-D(k3))
= (27r)3 63 (kl + k2 + k 3 )F(k l, k 2, k 3 ) .
(5.1)
The function F describes the correlation as a function of the triangle shape in momentum space.
The predictions for the function F in different models divide quite sharply into
two qualitatively different classes as a consequence of qualitatively different ways of
producing correlations among modes [4]. The first possibility is that the source of
density perturbations is not the inflaton but a second light scalar field a. In this case
non-Gaussianities are generated by the non-linear relation between the fluctuation
cr of this field and the final perturbation
1 we observe. This non-linearity is local
as it acts when the modes are much outside the horizon; schematically we have
4(x) = A a(x) + B(6U2(x)
-
(6a2)) +....
1
Even starting from a purely Gaussian
Even with perfectly Gaussian primordial fluctuations, the observables, e.g. the temperature
anisotropy, will not be perfectly Gaussian as a consequence of the non-linear relation between pri-5
mordial perturbations and what we will eventually observe. These effects are usually of order 10
(see for example [3]) and thus beyond (but not much) present sensitivity. In the following we will
disregard these contributions.
124
do, the quadratic piece introduces a 3-point function for
F(kl, k2, k 3 )= fl
2 4 (3
1
2
+
where A is the power spectrum normalization, ((kl)(
kl 3 ,
k3k
1
1
of the form
+
3)
k3 k2 k3
(5.2)
(k 2)) = (27r)3 53(kl + k 2 )A
-
which for the moment has been taken as exactly scale invariant, and where flL
is proportional to B. Examples of this mechanism are the curvaton scenario [5] and
the variable decay width model [6], which naturally give rise to fLal greater than 10
and 5, respectively.
The second class of models are single field models with a non-minimal Lagrangian,
where the correlation among modes is created by higher derivative operators [7, 8, 9,
10]. In this case, the correlation is strong among modes with comparable wavelength
and it decays when we take one of k's to zero keeping the other two fixed. Although
different models of this class give a different function F, all these functions are qualitatively very similar. We will call this kind of functions equilateral: as we will see,
the signal is maximal for equilateral configurations in Fourier space, whereas for the
local form (5.2) the most relevant configurations are the squeezed triangles with one
side much smaller than the others.
The strongest constraint on the level of non-Gaussianity comes from the WMAP
experiment. The collaboration analyzed their data searching for non-Gaussianity of
the local form (5.2), finding the data to be consistent with purely Gaussian statistics
and placing limits on the parameter
-58 < fal
f
al
[11]:
< 134 at 95% C.L.
(5.3)
The main purpose of the study in this chapter is to perform a similar analysis
searching for non-Gaussianities of the equilateral form. We can extend the definition
of f
'alin eq. (5.2) to a generic function F by setting the overall normalization on
equilateral configurations:
F(k, k, k) = fNL
125
6A2
O6
.k
(5.4)
In this way, two different models with the same fNL will give the same 3-point function
for equilateral configurations. For equilateral models in particular, the overall amplitude will be characterized by fquil.. Indirect constraints on fqUl. have been obtained
starting from the limit of eq. (5.3) in [4], resulting in fNqLUil.l 800; we will see that
a dedicated analysis is, as expected, much more sensitive. Many of the equilateral
models naturally predict sizeable values of the parameter fNLUi':ghost inflation and
DBI inflation [8, 9] tend to have fNLuil'
-
100, tilted ghost inflation fqUil. > 1 [10],
while slow roll inflation with higher derivative coupling typically give fNUil < 1 [7].
In Section 5.2 we discuss the main idea of the analysis. We will see that an optimal
analysis is numerically very challenging for a generic form of the function F, but
simplifies dramatically if the function F is factorizable in a sense that will be defined
below. As all the equilateral forms predicted in different models are (qualitatively and
quantitatively) very similar, our approach will be to choose a factorizable function
that well approximates this class. The analysis is further complicated by the breaking
of rotational invariance: a portion of the sky is fully masked by the presence of the
galaxy and moreover each point is observed a different number of times. In Section
5.3 we look for an optimal estimator for fNL. We discover that, unlike the rotationally
invariant case, the minimum variance estimator contains not only terms which are
cubic in the temperature fluctuations, but also linear pieces. These techniques are
used to analyze the WMAP data in Section 5.4. The result is that the data are
compatible with Gaussian statistics. The limit for the equilateral models is
-366 < fu.il < 238 at 95% C.L.
(5.5)
We also obtain a limit on fLal
-27 < fOCal < 121
at 95% C.L.
(5.6)
which is a slight improvement with respect to (5.3) as a consequence of the abovementioned additional linear piece in the estimator.
126
5.2
A factorizable equilateral shape
It is in principle straightforward to generalize the analysis for non-Gaussianities from
the local model to another one. We start assuming that the experimental noise on
the temperature maps is isotropic and that the entire sky is observed (no Galactic
and bright source mask); we will relax these assumptions in the next Section. In this
case it can be proved that an optimal estimator £ for fNL exists [12, 13], that is an
estimator which saturates the Cramer-Rao inequality and thus gives the strongest
possible constraint on the amplitude of the non-Gaussian signal. The estimator E
is a sum of terms cubic in the temperature fluctuations, each term weighted by its
signal to noise ratio:
.E
(5.7)
(allma 2m 2 a1 3m 3 ) alNmal2 m2 al3 m3
imC
1 C12Cl
all
12m2
3m
£
Given the assumptions, the power spectrum CI (which is the sum of the CMB signal
and noise) is diagonal in Fourier space; N is a normalization factor which makes
the estimator unbiased. From rotational invariance we can simplify the expression
introducing the Wigner 3j symbols to
I
£ =
11
'E
(
lN
imi
12
13
m2
m3
l
l
B11 12 13
1)
<
Cl111
a
a al
a
m
(5.8)
3 ,
(5.8)
where Bl 1 21 . is the angle-averaged bispectrum which contains all the information
about the model of non-Gaussianity we are considering. If B1112 13 is calculated for
fNL = 1, then the normalization factor N is given by
N=
(B112 3 )2
1l1213C, C2Ci
(5.9)
(59)
We now have to relate the angle-averaged bispectrum to the underlying correlation
among 3d modes F(kl, k 2, k 3). After some manipulations following [14], the estimator
127
takes the form
E=
(5.10)
00
N
S I d2
Ylml (n)Y
2m 2
()Y/
3m 3
r2 dr jll (klr)jl2(k 2 r)jl3(k 3 r) Cl C2 13
(n)
limi
0
J
2k32dk---k
2
kF(kl,
k2 , k3 )A (kl)
A T (k )AL
2
(k3 ) aLmal2m2alm3
where AT (k) is the CMB transfer function which relates the aim to the Newtonian
potential 1(k):
d 3k
aim = 47ri
T
(2)3
(k)J(k)Ylm(k).
(5.11)
Equation (5.10) is valid for any shape of the 3-point function F. Unfortunately this
expression is computationally very challenging. The sums over m can be taken inside
the integrals and factorized, but we are still left with a triple sum over I of an integral
over the sphere. The calculation time grows as N5 x2 ls where the number of pixels
Npixels
is of order 3 x 106 for WMAP. This approach is therefore numerically too
demanding.
As noted in [14], a crucial simplification is possible if the function F is factorizable
as a product of functions of k, k2 and k3 or can be written as a sum of a small
number of terms with this property. In this case the second line of (5.10) becomes
factored as the product of functions of each 1 separately, so that now also the sum
over can be done before integrating over the sphere. For example if we assume that
F(kl, k2, k3) = f(kl)f
-- '
=N
2 (k2)f 3 (k 3),
~J 2$f
dr
Jfr2dr
0
H1
|
i=1 limi
the estimator simplifiesto
Ym,(k).
1alam,
ji(kr)fi(k)T(k)C;
aimiimi(fn)
(5.12)
The calculation is obviously much faster now: it now scales like N3 2es and it is
dominated by going back and forth between real and spherical harmonics space.
From expression (5.2) we see that this simplification is possible for the local shape,
and it was indeed used for the analysis of the WMAP data in [11, ?].
128
Unfortunately, none of the "equilateral models" discussed in the Introduction
predicts a function F which is factorizable (see some explicit expressions in [4]),
so that it is not easy to perform an optimal analysis for a particular given model.
However, all these models give 3-point functions which are quite similar, so that it is
a very good approximation to take a factorizable shape function F which is close to
the class of functions we are interested in and perform the analysis for this shape. In
the limit k - 0 with k 2 and k 3 fixed, all the equilateral functions diverge as k- 1 [4]
(while the local form eq. (5.2) goes as k1 3 ). An example of a function which has this
behavior, is symmetric in k1, k2 and k 3, and is a sum of factorizable functions (and
is homogeneous of order k - 6 , see below) is
2
F(kl,k2, k3) = fNLU'-6A2
( kkk 2 k2
1 k3
k2
3
,
2
3
,2
3
+ (5perm.
(5.14)
where the permutations act only on the last term in parentheses. The mild divergence
for kl
-
0 is achieved through a cancellation among the various terms.
In figure 5-1, we compare this function with the local shape. The dependence
of both functions under a common rescaling of all k's is fixed to be oc k - 6 by scale
invariance, so that we can factor out k
6
for example. Everything will now depend
only on the ratios k 2 /kl and k 3 /k l , which fix the shape of the triangle in momentum
space. For each shape we plot F(1, k2 /kl, k 3 /kl)(k 2/k 1) 2 (k 3/kl) 2 ; this is the relevant
quantity if we are interested in the relative importance of different triangular shapes.
The square of this function in fact gives the signal to noise contribution of a particular
shape in momentum space [4]. We see that for the function (5.14), the signal to noise
2
Eq. (5.14) can be derived as follows. In order to make the divergence of F mild in the squeezed
limit we can use at the numerator a quantity which goes to zero in the same limit. The area of the
triangle does the job, going like k for kl - 0. The area can be expressed purely in terms of the
sides through Heron's formula [16], A = s(s - ki)(s - k 2 )(s - k3 ), where s = (kl
I +k 2 +k 3 ) is the
semiperimeter. The first s in the square root is irrelevant for our purposes, since it goes to a constant
in the squeezed limit; we will therefore omit it. Also we want a sum of factorizable functions, so we
get rid of the square root by considering A2 . In conclusion, a function with all the features stated
above is
(s - kl)(s - k )(s - k )
F(kwhich,
exactly
once
to
expanded,
eq.
(5.14).
reduces
2
which, once expanded, reduces exactly to eq. (5.14).
129
3
(5.13)
is concentrated on equilateral configurations, while squeezed triangles with one side
much smaller than the others are the most relevant for the local shape.
In figure 5-2 we study the equilateral function predicted both in the presence of
higher-derivative terms [7] and in DBI inflation [9]. In the second part of the figure
we show the difference between this function and the factorizable one used in our
analysis. We see that the relative difference is quite small. The same remains true
for other equilateral shapes (see [4] for the analogous plots for other models).
In [4], a "cosine" between different shapes was defined which quantifies how different is the signal given by two distributions.
The cosine is calculated from the
scalar product of the functions in fig. 5-1. We can think about this cosine as a sort of
correlation coefficient: if the cosine is close to 1 the two shapes are very difficult to
distinguish experimentally and an optimal analysis for one of them is very good also
for the other. On the other hand, a small cosine means that, once non-Gaussianities
are detected, there is a good chance to distinguish the two functions and that an
optimal analysis for one shape is not very good for the other. The cosine between
our template shape and the functions predicted by equilateral models is very close to
one (0.98 with the ghost inflation [8] prediction and 0.99 for higher derivative/DBI
models [7, 9]). This means that the error introduced in the analysis by the use of the
factorizable shape instead of the correct prediction for a given equilateral model is at
the percent level. On the other hand, as evident from figure 5-1, our template shape
is quite different from the local model - the cosine is merely 0.41.
All these numbers are obtained in 3 dimensions assuming that we can directly
measure the fluctuations with a 3d experiment like a galaxy survey.
The CMB
anisotropies are a complicated projection from the 3d modes. This makes it more
difficult to distinguish different shapes, although the picture remains qualitatively
the same.
Let us proceed with the study of our template shape. To further simplify the
130
Factorizable eq. shape
0.8
0.6
0.4
0.2
o
Local shape
6
4
2
1
0.8
Figure 5-1: Plot of the function F(l, k2/k1, k3/kl)(k2/kl?(k3/kl)2
for the equilateral
shape used in the analysis (top) and for the local shape (bottom). The functions are
both normalized to unity for equilateral configurations ~ = ~ = 1. Since F(k1, k2, k3)
is symmetric in its three arguments, it is sufficient to specify it for k1 2:: k2 2:: k3, so
~ ~ ~ ~ 1 above. Moreover, the triangle inequality says that no side can be longer than
the sum of the other two, so we only plot F in the triangular region 1 - ~ ~ ~ ~ ~ ~ 1
above, setting it to zero elsewhere.
131
estimator in eq. (5.10), we define the functions
a,(r)
(2)
-
2
+OO
2
r+O
2
r+OO
at(r)
2
T
dk
k
(5.15)
kr
A
dk k-l 1(k)j l (kr)A
+00
(5.16)
AT
dk/\ (k)jl(kr),
(5.18)
We use this strange ordering of the functions to keep the notation compatible with
[11, ?], where the functions a and p were introduced for the analysis of the local
shape. To evaluate
, we start from the spherical harmonic coefficients of the map
aim and calculate the four maps
A(r,h)- Zt atr)Yim(h)a'm
B(r, )
~Cl2
)Yjlm
(n)alm (519)
Im
C(r,f)=
/
Cl
Im
cYim(h)alm D(r, ) -
Im
(
()
Ylm()alm
(5.20)
Im
Now the estimator £ is given by
NJ2J
2h
£=-N8|
r2dr d2
nr
A(r, )B(r,
22
2+
)D( rn))r(rf)D(r'
(r, )D(r, n)j
)]
)
(5.21)
and the normalization N can be calculated from (5.9) using the explicit form
Bl l = :/(211+ 1)(212 + 1)(213 + 1)(11 0
0
12
0
13)
6 r2 dr [-a-,,(r)312(r)/311 (r) + (2 perm.)
-26 11 (r)612 (r)&13 (r) +
The functions a,
1l
(5.23)
(r)'Y12 (r)61 3 (r) + (5 perm.)]
, y and 6 used in the analysis can be obtained numerically start-
ing from the transfer functions A[(k),
which can be computed given a particu-
132
lar cosmological model with publicly available software as CMBFAST [17]. Some
plots of these functions are given in fig. 5-3, where we choose values of r close
to
0
-
(conformal time difference between recombination and the present), as
TR
these give the largest contribution to the estimator.
The oscillatory behavior in-
duced by the transfer function A (k) is evident, with a peak at 1
-
200. The
factors of 1(1+ 1) and (21 + 1) in the P and 6 functions, respectively, can be un-
derstood from the behavior at low l's, in the Sachs-Wolferegime. Here the transfer function can be approximated by AT(k) = -jl (k(0o that
fo
1(i(o
-
R) oc fo
0 dk k -1 j (k(ro -
dk j (k(ro -
TR))
R)) /3, and we then see
R)) oc 1/(1(1 + 1)) and
1(to - TR) oc
OC1/(21 + 1). From these expressions we also see that the
function 3 (the only one which is dimensionless) is very similar to the
the Sachs-Wolfe
regime,Cmb
5.3
- 2 0
dk k-' A j2(k(o -
Clmb's
as, in
TR)).
Optimal estimator for fNL
There are two quite general experimental complications that make the estimator £
defined in the last Section non-optimal. First, foreground emission from the Galactic
plane and from isolated bright sources contaminates a substantial fraction of sky,
which must be masked out before the analysis [18]. This projection which can be
accomplished by giving very large noise to regions that are affected by foregrounds,
besides reducing the available amount of data, has the important effect of breaking
rotational invariance, so that the signal covariance matrix
clm m12
2
of the masked
sky becomes non-diagonal in multipole space. Second, the noise level is not the same
in different regions of the sky, because the experiment looks at different regions for
different amounts of time. This implies that also the noise covariance matrix C1osel12
is non-diagonal in multipole space. The importance of the anisotropy of the noise was
pointed out in [11], as it caused an increase of the variance of the estimator £ at high
's.
For these reasons, it is worth trying to generalize the estimator £. We consider
estimators for fNL (for a given shape of non-Gaussianities) which contain both trilinear
133
and linear terms in the alm's. In the rotationally invariant case, a linear piece in the
estimator can only be proportional to the monopole, which is unobservable. However,
in the absence of rotational invariance, linear terms can be relevant. In this class, it
is straightforward to prove that the unbiased estimator with the smallest variance is
given by
Eii(a) =
((ama
2m 2
a
3
m 3 )i C11mI,14 m
2m 2,l 5m
4
5
13m3 ,6m6 al 4 m 4 alm 5
aAf4
)
limi
-3
1?
t~m
(allmlal2m2 al 3m 3)l Ci,12m2 C13 3 ,14 m4a4m 4 )
where N is a normalization factor:
N=
E
m C- 1
(alml al2 ml2 a 3m3 ) 1
llml,lm4
12m2,15m5
C-1
(a 4 m4 a 5 m5 al6 m6 )
(5.25)
lam3,16m6
limi
The averages (...)1 are taken with fNL = 1. As we stressed, the covariance matrix
C = Cc mb + Cnoise contains the effects of the mask projection and the anisotropic
noise and it is thus non-diagonal in multipole space.
It turns out that the estimator Elinsaturates the Cramer-Rao bound, i.e. it is
not only optimal in its class, but it is also the minimum variance unbiased estimator
among all the possible ones. To prove this, we expand the probability distribution
in the limit of weak non-Gaussianity (which is surely a good approximation for the
CMB) using the Edgeworth expansion [19, 20, 21].
P(afNL)
=
(1-fNL
a
a
a
e-2
0
E (alil al2m2ailm3) alml aal
2m2 aalm )ea1r)N
2
limi
13
144 l4m415m5
al5m5
V/(27r)Ndet(C)
(5.26)
Now, following the same arguments of [13]which considered the same problem in the
rotationally invariant case, the estimator is optimal (in the sense that its variance
saturates the Cramer-Rao bound), if and only if the following condition is satisfied:
dlogP(alfNL)= F(fNL)(in(a)- fNL) ,
d fNL
134
(5.27)
where F(fNL) is a generic function3 of the parameter fNL. From the probabil-
ity distribution (5.26) it is easy to check that the estimator
dlog P(afNL)/dfNL
Elin
is proportional to
in the limit of small fNL. We conclude that Olinis an optimal
estimator for a nearly Gaussian distribution.
We now want to make some approximations to the optimal estimator
Ei£,
to make
it numerically easier to evaluate. The full inversion of the covariance matrix is computationally rather cumbersome (although doable as the matrix is with good approximation block diagonal [22]). We therefore approximate C-1a in the trilinear term
of eq. (5.24) by masking out the sky before computing the alm's and taking C as
diagonal:
trilinear -a-l
where (a,,mal12 m 2 ala3 m)l
1
E
(ailmlal 2m2 al3 m3 )l
l
Ial,
12Mal_
,
2 imal m2am(.
12M2
,Ni0f1101203
limi
(5.28)
is still given by the full sky expressions of the last Section.
Once we have made this approximation for the trilinear term it is easy to prove that
the choice for the linear term which minimizes the variance is
linear
3
E
N limi
(allmlal
2 m 2 aa3 m 3
)
C11ml,l2m2 al 3 m 3
(5.29)
C11012013
Note that we must not approximate the covariance matrix C in the numerator as
diagonal. This would leave only a term proportional to the monopole a 00 (which is
unobservable), as in the rotationally invariant case.
The normalization factor N is given by
N=
Cl2Cl
fsky
C11213
(5.30)
where fsky is the fraction of the sky actually observed. As shown in [23], this correctly
takes into account the reduction of data introduced by the mask for multipoles much
higher than the inverse angular scale of the mask. The accuracy of this approximation
has been checked on non-Gaussian simulations for the local shape in [11].
3
When eq. (5.27) is satisfied, the function F turns out to be the Fisher information for the
parameter fNL, F(fNL) = ((dlog P(afNL)/dfNL) 2 ).
135
Let us try to understand qualitatively the effect of the linear correction. Take a
large region of the sky that has been observed many times so that its noise level is
low. This region will therefore have a small-scale power lower than average. Now,
in a given realization depending on how the large scale modes look like, this longwavelength modulation of the small-scale power spectrum may be "misinterpreted"
by the trilinear estimator as a non-Gaussian signal. Indeed, for the local shape, most
of the signal comes precisely from the correlation between long wavelength modes and
the small scale power [4]. On the other hand, for equilateral shapes, as we noted, the
signal is quite low on squeezed configurations so that we expect this effect to be small.
The linear term measures the correlation between a given map and the anisotropies
in the power spectrum, thus correcting for this spurious signal. Clearly the spurious
correlation is zero on average, but the effect increases the variance of the estimator.
We will apply the linear correction of the estimator only for the local shape, since,
as we will verify later, it gives a very small effect for equilateral shapes. Following
the same steps as in the last Section to factorize the trilinear term in the estimator,
we get an explicit expression for the linear piece in the local case:
3-
J r dr Jd f
2
2
E (2SAB(fn, r)
yl3,m3(fi) + SBB ('h, r) a1 (r)
a13 m 3 ,
Y 3 m3
13
13m3
(n)
(5.31)
where the two maps SAB(a, r) and
SBB(fi,
r) are defined as
SAB(,I,r) = E °Cl Yll,ml(~) C- Yl2,m2(f)(allmlal2m2)
(5.32)
limi
SBB(nfr) = E
limi
/
C11Ylr,ml1(n)
l'12
l12m2(n)(allmla
2m2)
(5.33)
As discussed above, it is only the anisotropic part of the matrix (allml a12 m 2 ) that gives
a contribution to the linear part of the estimator. This matrix can be decomposed as
(anoise noise
acmb)
(allmla2m2) = (acmb
allm1L=lm
2 +]-lmla2m2),
136
(534)
where ab
are the a's
of a map generated with CMB signal only and then per-
forming the mask projection, and amiseare the ones associated with a map generated
with noise only and then performing the mask projection. Both of these two matrices
have an anisotropic component. For the map (a mbal2mb2), this arises only because of
n ise), it is generated both by the sky cut and
noisea 12m2
the sky cut, while for the map \(allml
1
by the anisotropy of the noise power. We already discussed about the effect of the
anisotropy of the noise in the previous paragraph; this effect turns out to be the most
relevant. The sky cut gives a much smaller effect because, as we will explain more
in detail in the next Section, it is mainly associated with the average value of the
temperature outside of the sky cut, and this average value is subtracted out before
the analysis.
5.4
Analysis of WMAP 1-year data
We keep our methodology quite similar to the one used by the WMAP collaboration
for their analysis of the local shape [11] in order to have a useful consistency check.
We compare WMAP data with Gaussian Monte-Carlo realizations, which are used to
estimate the variance of the estimator. We generate with HEALPix 4 a random CMB
realization, with fixed cosmological parameters, at resolution nside=256 (786,432 pixels). The parameters are fixed to the WMAP best fit for a ACDM cosmology with
power-law spectrum [24]: Qbh2 = 0.024, Qmh2 = 0.14, h = 0.72, T = 0.17, n, = 0.99.
With these parameters, the present conformal time
bination time is
TR
T0
is 13.24 Gpc and the recom-
= 0.27 Gpc. A given realization is smoothed with the WMAP
window functions for the Q1, Q2, V1, V2, W1, W2, W3 and W4 bands [25]. To each
of these 8 maps we add an independent noise realization: for every pixel the noise
is a Gaussian random variable with variance Uo2/Nbs, where Nobs is the number of
observations of the pixel and u0 is a band dependent constant [26]. The maps are
then combined to give a single map: we make a pixel by pixel average of the 8 maps
weighted by the noise a2/Nobs We use this procedure because it is identical to that
4
See HEALPix website: http://www.eso.org/science/healpix/
137
used in [11], thereby allowing direct comparisons between our results and those of the
WMAP team. In future work, it can in principle be improved in two ways. First of
all, for the window function to be strictly rather than approximately uniform across
the sky, the weights of the eight input maps should be constant rather than variable
from pixel to pixel. This is a very small effect in practice, since the eight Nbs-maps
are very similar, making the weights close to constant.
Second, the sensitivity on
small scales can be improved by using -dependent weights [27]: for instance, at very
high 1, most of the weight should be given to the W bands, since they have the narrowest beam. This would also have a small effect for our particular application, since
our estimator uses only the first few hundred multipoles. We explicitly checked that
this would not reduce the variance of our estimator appreciably.
We apply the KpO mask to the final map, to cut out the Galactic plane and the
known point sources [18]: this mask leaves the 76.8% of the pixels, fsky = 0.768. The
average temperature outside the mask is then subtracted.
On the resulting map we calculate the estimators defined in the preceding Sections.
For the local shape we performed the analysis both with and without the linear
piece discussed in Section 5.3. The quadratic maps of equations (5.32) and (5.33)
used for the linear correction are calculated by averaging many (
600) Monte-Carlo
maps obtained with the same procedure above. We use HEALPix to generate and
analyze maps at resolution level nside=256 (786,432 pixels). The integration over r
is performed from
0
- 0.025 TR up to
0
- 2.5 TR with - 200 equidistant points, and
then with another logarithmically spaced - 60 points up to the present epoch. Such
a high resolution both in r spacing and in nside is necessary in order to reproduce
the cancellation on squeezed triangles which occurs among the various terms in the
equilateral shape (5.14). The computation of each fNL on a 2.0 GHz Opteron processor
with 2 GB of RAM takes about 60 minutes for the local shape, and 100 minutes for
the equilateral shape.
We then apply exactly the same procedure to WMAP data. These maps are analyzed after template foreground corrections are applied, in order to reduce foreground
signal leaking outside the mask, as described in [18].
138
A useful analytic bound on the variance of the estimators for flocal and feqil. is
obtained from the variance of the full sky estimator (with homogeneous noise) (5.8)
with an fsky-correction which takes into account the reduction in available information
from not observing the whole sky. Taking into account the normalization factor, one
readily obtains
D2
(5.35)
C1 C1 /21C3
an = fsky
11<12<13
where B11 213 must be evaluated for fcal
or fil.
equal to 1. As we discussed, our
approach is not strictly optimal as we are not inverting the full covariance matrix, so
we expect
Lan
to be smaller than the actual standard deviation that we measure from
our Monte-Carlos.
In figure 5-4, we show the standard deviation of estimators for the local shape
parameter fNal as a function of the maximum multipole analyzed.
We compare
the results of our Monte-Carlo simulations (with and without the linear correction)
with the analytic bound discussed above. The results without linear correction are
compatible with the analysis in [11]. We see that the addition of the linear piece
reduces the variance divergence at high l's.
The residual divergence is probably
associated with the fact that we did not invert the full covariance matrix.
estimator with the smallest variance is the one with linear correction at
max
The
= 335,
with a standard deviation of 37. The analytic approximation has an asymptotic
value of 30 which should be considered the best possible limit with the present data.
Our estimator thus extracts about (37/30)- 2
66% of the fal-information
(inverse
variance) from the WMAP data. The analysis of the data with lmax = 335 gives 47,
so there is no evidence of deviation from pure Gaussianity and we can set a 2a limit
of
-27 < fINL < 121 at 95% C.L.
(5.36)
In fig. 5-5, we show a map SAB(n, r) for a radius around recombination calculated
with noise only and for Imax= 370, as an illustrative example of the role of the linear
piece. As we will clarify below, it is in fact the anisotropy of the noise that causes most
of the contribution to the linear piece. The companion map SBB(n, r) is qualitatively
139
very similar to the map SAB. It is the anisotropy of the noise that gives rise to a nontrivial (i.e. non-uniform) SAB map, as is clear from the comparison in the same figure
with a plot of the number of observations per pixel. This can help us in understanding
the contribution of the linear piece associated with the anisotropy of the noise to the
estimator. Let us consider a particular Gaussian Monte Carlo realization with a long
wavelength mode crossing one of the regions with a high number of observations.
The trilinear piece of the estimator will then detect a spurious non-Gaussian signal
associated with the correlation between this long wavelength mode and the small short
scale power of the noise (5). The maps SAB and SBB, because of their particular shape,
will have a non-zero dot product with precisely the same long wavelength mode of the
Monte Carlo map, and with an amplitude proportional to the anisotropy of the two
point function of the noise, thus effectively subtracting the spurious signal detected
by the trilinear piece and reducing the variance of the estimator.
As the mask breaks rotational invariance, the CMB signal also gives non-uniform
S-maps. However, the breaking of rotational invariance can be neglected far from the
galaxy mask, so we expect that the contribution to the S-maps from the CMB is to
first approximation constant outside the mask (and zero inside). In this approximation, the linear contribution of the estimator would be sensitive only to the average
value outside the mask. However, given that we are constraining the statistical properties of temperature fluctuations, the average temperature in the region outside the
mask has to be subtracted before performing the analysis; therefore we expect the
linear correction associated with the CMB signal to be rather small. These expectations are in fact verified by our simulations. We checked that the maps S coming from
the CMB signal are to first approximation constant outside the mask, with additional
features coming from the patches used to mask out bright sources outside the Galactic
plane. We then verified that if the average value of the temperature is not subtracted,
the linear term coming from the CMB signal gives a very important reduction of the
estimator variance, while its effect is almost negligible when the mean temperature is
5
The effect of this spurious signal obviously averages to zero among many Monte Carlo realizations
but it increases the variance of the estimator.
140
subtracted, as we do in the analysis.
Uil . The
In figure 5-6, we study the standard deviation of the estimators for fNL
estimator without linear corrections is quite close to the information-theoretical limit,
so we did not add the linear corrections (which would require many averages analogous
to the maps SAB and SBB). The reason why we have such a good behavior is that,
as mentioned, here most of the signal comes from equilateral configurations. Thus
the estimator is not very sensitive to the correlation between the short scale power
and the (long wavelength) number of observations.
Moreover the inversion of the
covariance matrix is important only for the low multipoles, which give only a small
fraction of the signal for fLil
'.
We find that the estimator with smallest standard
deviation is at m,, = 405, with a standard deviation of 151. The analysis of the data
with the same Im,, gives -64. We deduce a 2a limit
-366 < fL u". < 238
at 95% C.L.
(5.37)
The given limit is approximately 2 times stronger than the limit obtained in [4],
indirectly obtained starting from the limit on f
al. The limit appears weaker than for
the local case because, for the same fNL, the local distribution has larger signal to noise
ratio than an equilateral one, as evident from fig. 5-1. This is not a physical difference
but merely a consequence of our defining fNL at the equilateral configuration.
To check that the two limits correspond approximately to the same "level of
non-Gaussianity" we can define a quantity which is sensitive to the 3-point function
integrated over all possible shapes of the triangle in momentum space. In this way
it will be independent of which point we choose for normalization.
We define this
quantity, which we will call NG, directly in 3d as an integral of the square of the
functions in fig. 5-1
NG
(/E
(
X2dzX2 3dx
2 he
3
F(1, X2,
2
3)
o Aie Xi-3
1/2
(5.38)
(5.38)
The integration is restricted to the same triangular region as in fig.s 5-1 and 5-2:
141
1-
2
_< X 3
<
X2
<
1. The measure of integration comes from the change of
variables from the 3d wavevectors to the ratios x2 = k 2 /k1 and
parametrically of order
A/ 2
X3
= k 3 /kl.
NG is
· fNL (where A - 1.9.10 - 8 [24, 28]), which is the correct
order of magnitude of the non-Gaussianity, analogous to the skewness for a single
random variable. To better understand the meaning of the defined quantity, note
that if we integrate inside the parentheses in eq. (5.38) over the remaining wavevector
we get the total signal to noise ratio (or better the non-Gaussian to Gaussian ratio).
This further integration would approximately multiply NG by the square root of the
number of data. This means that to detect a given value of NG we need a number of
data of order NG - 2 , in order to have a signal to noise ratio of order 1. This is clearly
a good way to quantify the deviation of the statistics from pure Gaussianity. The 2c
windows for fNL can be converted to constraints on NG (we define NG to have the
same sign of
fNL) 6
-0.006 < NGlOCal< 0.025 at 95% C.L.
(5.39)
at 95% C.L.
(5.40)
-0.016 < NGeqUil.< 0.010
Contrary to what one might think from a naive look at the limits on fNL,the maximum
tolerated amount of non-Gaussian signal in a map is thus very similar for both shapes.
So far in this chapter, we have neglected any possible dependence on the cosmological parameters.
A proper analysis would marginalize over our uncertainties in
parameters and this would increase the allowed range of fNL. In order to estimate
what the effect of these uncertainties is, let us imagine that the real cosmological
parameters are not exactly equal to the best fit ones. The cosmological parameters
are determined mainly from the 2-point function of the CMB, Clmb, therefore the
largest error bars are associated to those combinations of parameters which leave
Clmb
unchanged. In the limit in which the Cl
mb's
remain the same, also the variance
of our estimator (which is computed with the best fit cosmological parameters for
6
Note that for the local model the integration in eq. (5.38) diverges like the log of the ratio of the
minimum to the maximum scales in the integral. For the quoted numbers we chose a ratio of 1000.
142
a ACDM cosmology with power-law spectrum [24]) remains the same: in the weak
non-Gaussian limit the variance just depends on the 2-point function. However, the
combination of parameters which leave unchanged the Clmb's does not necessarily
leave unchanged the bispectrum, therefore uncertainties in the determination of the
cosmological parameters will have an influence on the expectation value of the esti-
mator. The expectation value of the estimator would be:
() =
1
1
N =j
(5.41)
11C12B13
0
C1C12Ct3
where N, Bht213a and C1 are the same as in our analysis, computed with the best
fit cosmological parameters, while Bl,1213 is the true bispectrum. Thus when varying
parameters, the normalization N needs to be changed to make the estimator unbiased.
The most relevant uncertainty is the reionization optical depth
, which is cor-
related with the uncertainty in the amplitude of the power spectrum &A. With the
purpose of having a rough estimate, we can approximate the effect of reionization by
a multiplicative factor e -T in front of the transfer function A\'(k) for 1 corresponding to scales smaller than the horizon at reionization. In order to keep the Clfmb's
unchanged, at least at high l's, we then multiply A by e2 r. The bispectrum is proportional to A2 A(k) 3 and thus changes even if the
Cl's
do not. For the equilateral
shape most of the signal comes from equilateral configurations with all the 3 modes
inside the horizon at reionization; in this case () scales roughly as e. For the local
shape the signal comes from squeezed configurations with one mode of much smaller
wavelength than the others. Taking only 2 modes inside the horizon at reionization
we get ()
oc e2 .
If we consider the la error, r = 0.166+0.076[24], we find that the uncertainty
in the reionization depth should correspond to an error on fNL of order 8% for the
equilateral shape, and 15% for the local shape. These numbers translate directly into
uncertainties in the allowed ranges we quote. In the future, if the error induced by the
uncertainty in the cosmological parameters will become comparable to the variance
of the estimator, a more detailed analysis will be required; however, at the moment
143
an analysis with fixed cosmological parameters is certainly good enough.
5.5
Conclusions
We have developed a method to constrain the level of non-Gaussianity when the
induced 3-point function is of the "equilateral" type. We showed that the induced
shape of the 3-point function can be very well approximated by a factorizable form
making the analysis practical. Applying our technique to the WMAP first year data
we obtained
-366 < fHq"l. < 238 at 95% C.L.
(5.42)
The natural expectation for this amplitude for ghost or DBI inflation is fuil.
100,
below the current constraints but at a level that should be attainable in the future.
The limit an experiment can set on fNL just scales as the inverse of the maximum
the experiment can detect. As a result, in the case of WMAP, increased observing
time will approximately decrease the error bars by 30% and 60% for 4 years and 8
years of observation. The increased angular resolution and smaller noise of Planck
pushes the point where noise dominates over signal to 1 - 1500. This should result
in a factor of 4 improvement on the present constraints.
In addition polarization
measurements by Planck can further reduce the range by an additional factor of 1.6
[29].
We also constrained the presence of a 3-point function of the "local" type, predicted for example by the curvaton and variable decay width inflation models, ob-
taining
-27 < fNL < 121 at 95% C.L.
(5.43)
We defined a quantity NG, which quantifies for any shape the level of nonGaussianity of the data, analogously to the skewness for a single random variable.
The limits on NG are very similar for the two shapes, approximately NG] < 0.02 at
95% C.L. .
We showed that unless one has a full sky map with uniform noise, the estimator
144
must contain a piece that is linear on the data in order to extract all the relevant
information from the data and saturate the Cramer-Rao bound for the 3-point function measurement uncertainty. This correction is particularly important for the local
shape and accounts for the improvement of our limits over that from the WMAP
team. Moreover this correction goes a long way towards reducing the divergence in
the variance of the estimator as Ima is increased.
145
Higher derivative
0.8
0.6
0.4
0.2
o
1
Difference
0.8
0.6
0.4
0.2
o
1
0.8
0.6
Figure 5-2: Top. Plot of the function F(l, k2/kl, k3/kJ)(k2/kl?(k3/kl?
predicted by the
higher-derivative [7] and the DBI models [9]. Bottom. Difference between the above plot
and the analogous one (top of fig. 5-1) for the factorizable equilateral shape used in the
analysis.
146
1(1+1)P x 109
ax 1011
4F
I
-1
-/ -
-
'%
2
100
200
300
.---------)-?
.
1'
1
-2
:1
-2
-4
y x 1012
,''
(21+1) x 1010
3
-I
4
'I
200
12
/
100
200
300
,
/ ,'100
200_
/
-2
- i
ro- 0.7TR--.--.
-3-3
-8
r-70
To - 1.6 TR -1
To -1.3
TR -To -TR
o - 0.
4
TR
Figure 5-3: The functions al(r) (in units of Mpc-3 ), /31(r) (dimensionless), yl(r) (in units
of Mpc-2), and 61(r) (in units of Mpc -1 ) are shown for various radii r as functions of the
multipole number 1. The cosmological parameters are the same ones used in the analysis:
2
Q2bh
= 0.024, Qmh2 = 0.14, h = 0.72, r = 0.17. With these parameters, the present
conformal time ' 0 is 13.24 Gpc and the recombination time
147
TR
is 0.27 Gpc.
Aflocal
7n N L
Iu
60
f
I
50
40
30
20
i
i
4
10
··
I .
250
. . .
. .
300
.
350
400
Imax
Figure 5-4: Standard deviation for estimators of flmal as a function of the maximum
used in the analysis. Lower curve: lower bound deduced from the full sky variance. Lower
data points: standard deviations for the trilinear + linear estimator. Upper data points:
the same for the estimator without linear term, for which the divergence at high 's caused
by noise anisotropy had already been noticed in [11].
148
Figure 5-5: Top: SAB(n, r) map for r around recombination calculated with noise only.
Bottom: number of observations as a function of the position in the sky for the Ql band
(the plot is quite similar for the other bands). A lighter color indicates points which are
observed many times.
149
Afe q
250
200
f
150
i
f~
100
50
250
300
350
400
450 Ima
Figure 5-6: Standard deviations for estimators of fjNL. as a function of the maximum 1
used in the analysis. Lower curve: lower bound deduced from the full sky variance. Data
points: standard deviations for the estimator without linear term.
150
Bibliography
[1] J. Maldacena, "Non-Gaussian features of primordial fluctuations in single field
inflationary models," JHEP 0305, 013 (2003) [astro-ph/0210603].
[2] V. Acquaviva, N. Bartolo, S. Matarrese and A. Riotto, "Second-order cosmological perturbations from inflation," Nucl. Phys. B 667, 119 (2003) [astroph/0209156].
[3] P.
"CMB 3-point functions generated by
Creminelli and M. Zaldarriaga,
non-linearities at recombination,"
Phys. Rev. D 70, 083532 (2004) [astro-
ph/0405428].
[4] D. Babich, P. Creminelli and M. Zaldarriaga, "The shape of non-Gaussianities,"
JCAP 0408, 009 (2004) [astro-ph/0405356].
[5] D. H. Lyth, C. Ungarelli and D. Wands, "The primordial density perturbation
in the curvaton scenario," Phys. Rev. D 67, 023503 (2003) [astro-ph/0208055].
[6] M. Zaldarriaga,
"Non-Gaussianities in models with a varying inflaton decay
rate," Phys. Rev. D 69, 043508 (2004) [astro-ph/0306006].
[7] P. Creminelli, "On non-Gaussianities in single-field inflation," JCAP 0310, 003
(2003) [astro-ph/0306122].
[8] N. Arkani-Hamed, P. Creminelli, S. Mukohyama and M. Zaldarriaga, "Ghost
inflation," JCAP 0404, 001 (2004) [hep-th/0312100].
[9] M. Alishahiha, E. Silverstein and D. Tong, "DBI in the sky," Phys. Rev. D 70,
123505 (2004) [hep-th/0404084].
151
[10] L. Senatore, "Tilted ghost inflation," Phys. Rev. D 71, 043512 (2005) [astroph/0406187].
[11] E. Komatsu et al., "First Year Wilkinson Microwave Anisotropy Probe (WMAP)
Observations: Tests of Gaussianity," Astrophys. J. Suppl. 148, 119 (2003) [astroph/0302223].
[12] A. F. Heavens, "Estimating non-Gaussianity in the microwave background,"
astro-ph/9804222.
[13] D. Babich, "Optimal Estimation of Non-Gaussianity," astro-ph/0503375.
[14] L. M. Wang and M. Kamionkowski, "The cosmic microwave background bispectrum and inflation," Phys. Rev. D 61, 063504 (2000) [astro-ph/9907431].
[15] E. Komatsu, D. N. Spergel and B. D. Wandelt, "Measuring primordial nonGaussianity in the cosmic microwave background," astro-ph/0305189.
[16] Heron of Alexandria, Metrica, Alexandria, circa 100 B.C. - 100 A.D.
[17] U. Seljak and M. Zaldarriaga, "A Line of Sight Approach to Cosmic Microwave
Background Anisotropies," Astrophys. J. 469, 437 (1996) [astro-ph/9603033].
[18] C. Bennett et al., "First Year Wilkinson Microwave Anisotropy Probe (WMAP)
Observations: Foreground Emission," Astrophys. J. Suppl. 148, 97 (2003) [astroph/0302208].
[19] R. Juszkiewicz, D. H. Weinberg, P. Amsterdamski,
M. Chodorowski and
F. Bouchet, "Weakly nonlinear Gaussian fluctuations and the Edgeworth expansion," Astrophys. J. 442 (1995) 39.
[20] A. Taylor and P. Watts, "Parameter information from nonlinear cosmological
fields," Mon. Not. Roy. Astron. Soc. 328, 1027 (2001) [astro-ph/0010014].
[21] F. Bernardeau and L. Kofman, "Properties of the Cosmological Density Distribution Function," Astrophys. J. 443 (1995) 479 [astro-ph/9403028].
152
[22] G. Hinshaw et al., "First Year Wilkinson Microwave Anisotropy Probe (WMAP)
Observations: Angular Power Spectrum," Astrophys. J. Suppl. 148, 135 (2003)
[astro-ph/0302217].
[23] E. Komatsu, "The Pursuit of Non-Gaussian Fluctuations in the Cosmic Microwave Background," astro-ph/0206039.
[24] D. N. Spergel et al. [WMAP Collaboration], "First Year Wilkinson Microwave
Anisotropy Probe (WMAP) Observations: Determination of Cosmological Parameters," Astrophys. J. Suppl. 148, 175 (2003) [astro-ph/0302209].
[25] L. Page et al., "First Year Wilkinson Microwave Anisotropy Probe (WMAP)
Observations: Beam Profiles and Window Functions," Astrophys. J. Suppl. 148,
39 (2003) [astro-ph/0302214].
[26] C. L. Bennett
et al., "First Year Wilkinson Microwave Anisotropy Probe
(WMAP) Observations: Preliminary Maps and Basic Results," Astrophys. J.
Suppl. 148, 1 (2003) [astro-ph/0302207].
[27] M. Tegmark, A. de Oliveira-Costa & A.J.S. Hamilton, "A high resolution foreground cleaned CMB map from WMAP", Phys. Rev. D 68, 123523 (2003) [astroph/0302496].
[28] L. Verde et al., "First Year Wilkinson Microwave Anisotropy Probe (WMAP)
Observations: Parameter Estimation Methodology," Astrophys. J. Suppl. 148,
195 (2003) [arXiv:astro-ph/0302218].
[29] D. Babich and M. Zaldarriaga, "Primordial Bispectrum Information from CMB
Polarization," Phys. Rev. D 70, 083005 (2004) [astro-ph/0408455].
[30] K. M. Gorski, E. Hivon and B. D. Wandelt, "Analysis Issues for Large CMB
Data Sets," astro-ph/9812350.
153
Chapter 6
The Minimal Model for Dark
Matter and Unification
Gauge coupling unification and the success of TeV-scale weakly interacting dark matter are usually taken as evidence of low energy supersymmetry (SUSY). However, if
we assume that the tuning of the higgs can be explained in some unnatural way, from
environmental considerations for example, SUSY is no longer a necessary component
of any Beyond the Standard Model theory. In this chapter we study the minimal
model with a dark matter candidate and gauge coupling unification. This consists of
the SM plus fermions with the quantum numbers of SUSY higgsinos, and a singlet.
It predicts thermal dark matter with a mass that can range from 100 GeV to around
2 TeV and generically gives rise to an electric dipole moment (EDM) that is just
beyond current experimental limits, with a large portion of its allowed parameter
space accessible to next generation EDM and direct detection experiments. We study
precision unification in this model by embedding it in a 5-D orbifold GUT where
certain large threshold corrections are calculable, achieving gauge coupling and b-T
unification, and predicting a rate of proton decay just beyond current limits.
154
6.1
Introduction
Over the last few decades the search for physics beyond the Standard Model (SM) has
largely been driven by the principle of naturalness, according to which the parameters
of a low energy effective field theory like the SM should not be much smaller than the
contributions that come from running them up to the cutoff. This principle can be
used to constrain the couplings of the effective theory with positive mass dimension,
which have a strong dependence on UV physics. Requiring no fine tuning between
bare parameters and the corrections they receive from renormalization means that
the theory must have a low cutoff. New physics can enter at this scale to literally cut
off the high-energy contributions from renormalization.
In the specific case of the SM the effective lagrangian contains two relevant parameters: the higgs mass and the cosmological constant (c.c), both of which give rise
to problems concerning the interpretation of the low energy theory. Any discussion
of large discrepancies between expectation and observation must begin with what
is known as the c.c. problem. This relates to our failure to find a well-motivated
dynamical explanation for the factor of 10120between the observed c.c and the naive
contribution to it from renormalization which is proportional to A4 , where A is the
cutoff of the theory, usually taken to be equal to the Planck scale. Until very recently
there was still hope in the high energy physics community that the c.c. might be
set equal to zero by some mysterious symmetry of quantum gravity. This possibility
has become increasingly unlikely with time since the observation that our universe is
accelerating strongly suggests the presence of a non-zero cosmological constant [1, 2].
A less extreme example is the hierarchy between the higgs mass and the GUT
scale which can be explained by SUSY breaking at around a TeV. Unfortunately the
failure of indirect searches to find light SUSY partners has brought this possibility
into question, since it implies the presence of some small fine-tuning in the SUSY
sector. This 'little hierarchy' problem [3, 4] raises some doubts about the plausibility
of low energy SUSY as an explanation for the smallness of the higgs mass.
Both these problems can be understood from a different perspective: the fact
155
that the c.c. and the higgs mass are relevant parameters means that they dominate
low energy physics, allowing them to determine very gross properties of the effective
theory. We might therefore be able to put limits on them by requiring that this theory
satisfy the environmental conditions necessary for the universe not to be empty. This
approach was first used by Weinberg [5]to deduce an upper bound on the cosmological
constant from structure formation, and was later employed to solve the hierarchy
problem in an analogous way by invoking the atomic principle [6].
Potential motivation for this class of argument can be found in the string theory
landscape. At low energies some regions of the landscape can be thought of as a field
theory with many vacua, each having different physical properties. It is possible to
imagine that all these vacua might have been equally populated in the early universe,
but observers can evolve only in the few where the low energy conditions are conducive
to life. The number of vacua with this property can be such a small proportion of the
total as to dwarf even the tuning involved in the c.c. problem; resolving the hierarchy
problem similarly needs no further assumptions. This mechanism for dealing with
both issues simultaneously by scanning all relevant parameters of the low energy
theory within a landscape was recently proposed in [7, 8].
From this point of view there seems to be no fundamental inconsistency with
having the SM be the complete theory of our world up to the Planck scale; nevertheless
this scenario presents various problems. Firstly there is increasing evidence for dark
matter (DM) in the universe, and current cosmological observations fit well with the
presence of a stable weakly interacting particle at around the TeV scale. The SM
contains no such particle. Secondly, from a more aesthetic viewpoint gauge couplings
do not quite unify at high energies in the SM alone; adding weakly interacting particles
changes the running so unification works better. A well-motivated example of a model
that does this is Split Supersymmetry [7], which is however not the simplest possible
theory of this type. In light of this we study the minimal model with a finely-tuned
higgs and a good thermal dark matter candidate proposed in [8], which also allows
for gauge coupling unification. Although a systematic analysis of the complete set of
such models was carried out in [9], the simplest one we study here was missed because
156
the authors did not consider the possibility of having large UV threshold corrections
that fix unification, as well as a GUT mechanism suppressing proton decay.
Adding just two 'higgsino' doublets' to the SM improves unification significantly.
This model is highly constrained since it contains only one new parameter, a Dirac
mass term for the doublets ('pu'), the neutral components of which make ideal DM
candidates
for 990 GeV<
L < 1150 GeV (see [9] for details).
However a model
with pure higgsino dark matter is excluded by direct detection experiments since
the degenerate neutralinos have unsuppressed vector-like couplings to the Z boson,
giving rise to a spin-independent direct detection cross-section that is 2-3 orders of
magnitude above current limits2 [10, 11]. To circumvent this problem, it suffices to
include a singlet ('bino') at some relatively high energy (
109 GeV), with yukawa
couplings with the higgsinos and higgs, to lift the mass degeneracy between the 'LSP'
and 'NLSP' 3 by order 100 keV [12], as explained in Appendix 6.8. The instability of
such a large mass splitting between the higgsinos and bino to radiative corrections,
which tend to make the higgsinos as heavy as the bino, leads us to consider these
masses to be separated by at most two orders of magnitude, which is technically
natural. We will see that the yukawa interactions allow the DM candidate to be as
heavy as 2.2 TeV. There is also a single reparametrization invariant CP violating
phase which gives rise to a two-loop contribution to the electron EDM that is well
within the reach of next-generation experiments.
Our chapter is organized as follows: in Section 6.2 we briefly introduce the model,
in Section 6.3 we study the DM relic density in different regions of our parameter space
with a view to constraining these parameters; we look more closely at the experimental
implications of this model in the context of dark matter direct detection and EDM
experiments in Sections 6.4 and 6.5. Next we study gauge coupling unification at
two loops. We find that this is consistent modulo unknown UV threshold corrections,
1
Here 'higgsino' is just a mnemonic for their quantum numbers, as these particles have nothing
to do with the SUSY partners of the higgs.
2
A model obtained adding a single higgsino doublet, although more minimal, is anomalous and
hence is not considered here.
3
From here on we will refer to these particles and couplings by their SUSY equivalents without
the quotation marks for simplicity.
157
however the unification scale is too low to embed this model in a simple 4D GUT.
This is not necessarily a disadvantage since 4D GUTs have problems of their own, in
splitting the higgs doublet and triplet for example. A particularly appealing way to
solve all these problems is by embedding our model in a 5D orbifold GUT, in which
we can calculate all large threshold corrections and achieve unification. We also find a
particular model with b-r unification and a proton lifetime just above current bounds.
We conclude in Section 6.7.
6.2
The Model
As mentioned above, the model we study consists of the SM with the addition of two
fermion doublets with the quantum numbers of SUSY higgsinos, plus a singlet bino,
with the following renormalizable interaction terms:
1
I'JuTd + 1 MI'T
2
where Is is the bino,
'u,d
s + AuIuThJs + AdAdhit s
(6.1)
are the higgsinos, h is the finely-tuned higgs.
We forbid all other renormalizable couplings to SM fields by imposing a parity
symmetry under which our additional particles are odd whereas all SM fields are even.
As in SUSY conservation of this parity symmetry implies that our LSP is stable.
The size of the yukawa couplings between the new fermions and the higgs are
limited by requiring perturbativity to the cutoff. For equal yukawas this constrains
A(Mz)
= Ad(Mz)
0.88, while if we take one of the couplings to be small, say
Ad(Mz) = 0.1 then Au(Mz) can be as large as 1.38.
The above couplings allow for the CP violating phase 0 = Arg(MlA*\A),
giving
5 free parameters in total. In spite of its similarity to the MSSM (and Split SUSY)
weak-ino sector, there are a number of important differences which have a qualitative
effect on the phenomenology of the model, especially from the perspective of the relic
density. Firstly a bino-like LSP, which usually mixes with the wino, will generically
annihilate less effectively in this model since the wino is absent. Secondly the new
158
yukawa couplings are free parameters so they can get much larger than in Split SUSY,
where the usual relation to gauge couplings is imposed at the high SUSY breaking
scale. This will play a crucial role in the relic density calculation since larger yukawas
means greater mixing in the neutralino sector as well as more efficient annihilation,
especially for the bino which is a gauge singlet.
Our 3x3 neutralino mass matrix is shown below:
MN=
M1
Auv
AdV
A,,v
0
-- ie i
Adv
-Ae io
0
for v = 174 GeV, where we have chosen to put the CP violating phase in the /z term.
The chargino is the charged component of the higgsino with tree level mass A.
It is possible to get a feel for the behavior of this matrix by diagonalizing it
perturbatively for small off-diagonal terms, this is done in Appendix 6.8.
6.3
Relic Abundance
In this section we study the regions of parameter space in which the DM abundance
is in accordance the post-WMAP 2a region 0.094 < Qdmh2 < 0.129 [2], where RQdm
is the fraction of today's critical density in DM, and h = 0.72 ± 0.05 is the Hubble
constant in units of 100 km/(s Mpc).
As in Split SUSY, the absence of sleptons in our model greatly decreases the
number of decay channels available to the LSP [13, 14]. Also similar to Split SUSY is
the fact that our higgs can be heavier than in the MSSM (in our case the higgs mass is
actually a free parameter), hence new decay channels will be available to it, resulting
in a large enhancement of its width especially near the WW and ZZ thresholds.
This in turn makes accessible neutralino annihilation via a resonant higgs, decreasing
the relic density in regions of the parameter space where this channel is accessible.
For a very bino-like LSP this is easily the dominant annihilation channel, allowing
the bino density to decrease to an acceptable level. We use a modified version of
159
the DarkSUSY [15] code for our relic abundance calculations, explicitly adding the
resonant decay of the heavy higgs to W and Z pairs.
As mentioned in the previous section there are also some differences between our
model and Split SUSY that are relevant to this discussion: the first is that the Minimal
Model contains no wino equivalent (this feature also distinguishes this model from
that in [16], which contains a similar dark matter analysis). The second difference
concerns the size of the yukawa couplings which govern this mixing, as well as the
annihilation cross-sectionto higgses. Rather than being tied to the gauge couplings
at the SUSY breaking scale, these couplings are limited only by the constraint of
perturbativity
to the cutoff. This means that the yukawas can be much larger in
our model, helping a bino-like LSP to both mix more and annihilate more efficiently.
These effects are evident in our results and will be discussed in more detail below.
We will restrict our study of DM relic abundance and direct detection in this
model to the case with no CP violating phase ( = 0, 7r); we briefly comment on the
general case in Section 6.5. Our results for different values of the yukawa couplings
are shown in Figure 6-1 below, in which we highlight the points in the t-M1 plane
that give rise to a relic density within the cosmological bound. The higgs is relatively
heavy (Mhiggs= 160 GeV) in this plot in order to access processes with resonant
annihilation through an s-channel higgs. As we will explain below the only effect this
has is to allow a low mass region for a bino-like LSP with M1l
Mhiggs/2.
Notice that the relic abundance seems to be consistent with a dark matter mass
as large as 2.2 TeV. Although a detailed analysis of the LHC signature of this model
is not within the scope of this study, it is clear that a large part of this parameter
space will be inaccessible at LHC. The pure higgsino region for example, will clearly
be hard to explore since the higgsinos are heavy and also very degenerate. There is
more hope in the bino LSP region for a light enough spectrum.
While analyzing these results we must keep in mind that
Qdm
10- 9GeV-2/(a)eff,
where ()eff is an effective annihilation cross section for the LSP at the freeze out
temperature, which takes into consideration all coannihilation channels as well as the
thermal average [17]. It will be useful to approximate this quantity as the cross160
3000
2000
I
....
..
annihilates too much
annihilates too little
.....
1000
...
500
1000
+
(}=O, >'u=O.I, >'d=O.1
•
(}=7r, >'u=O.I, >'d=O.1
•
(}=O, >'u=0.88, >'d=0.88
•
(}=7r, >.,,=0.88, >'d=0.88
•
(}=O, >'u=1.38,
•
(}=7r, >',,=1.38, >'d=O.1
2000
1500
>'d=O.1
2500
3000
/l (GeV)
Figure 6-1: Graph showing regions of parameter space consistent with WMAP.
section for the dominant
annihilation
will help us build some intuition
channel.
Although rough, this approximation
on the behavior
of the relic density in different
parts of the parameter
space.
We will not discuss the region close to the origin
where the interpretation
of the results become more involved due to large mixing and
coannihilation.
6.3.1
Higgsino Dark Matter
In order to get a feeling for the structure
of Figure 6-1, it is useful to begin by
looking at the regions in which the physics is most simple. This can be achieved by
diminishing the the number of annihilation
channels that are available to the LSP by
taking the limit of small yukawa couplings.
For B
imation
=
0, mixing occurs only on the diagonal M1
(see Appendix
=
Jl,
to a very good approx-
6.8 and Figure 6-2), hence the region above the diagonal
corresponds
to a pure higgsino LSP with mass J-L. For Au = Ad = 0.1 the yukawa
interactions
are irrelevant and the LSP dominantly
annihilates
by t-channel
neutral
(charged) higgsino exchange to Z Z (WW) pairs. Charginos, which have a tree-level
161
Ann
200(
2000
100(
u
1000
1
IVi
7MnN
----
I
.
I
'
.
.
.
.
'---
..
1f000
.
....
----
Afl
- -- -
I (GeV)
S (GeV)
(b) 0=7r
(a) 0=0
Figure
.
6-2: Gaugino fraction contours for A, =
Ad
= 0.88 and 0=0 (left),7r (right).
mass /u and are almost degenerate with the LSP, coannihilate with it, decreasing the
relic density by a factor of 3. This fixes the LSP mass to be around A= 1 TeV, giving
rise to the wide vertical band that can be seen in the figure; for smaller /u the LSP
over-annihilates, for larger /i it does not annihilate enough.
Increasing the yukawa couplings increases the importance of t-channel bino exchange to higgs pairs. Notice that taking the limit M1 >> u makes this new interaction
irrelevant, therefore the allowed region converges to the one in which only gauge interactions are effective. Taking this as our starting point, as we approach the diagonal
the mass of the bino decreases, causing the t-channel bino exchange process to become
less suppressed and increasing the total annihilation cross-section. This explains the
shift to higher masses, which is more pronounced for larger yukawas as expected and
peaks along the diagonal where the higgsino and bino are degenerate and the bino
mass suppression is minimal. The increased coannihilation between higgsinos and
binos close to the diagonal does not play a large part here since both particles have
access to a similar t-channel diagram.
Taking 0 = r makes little qualitative difference when either of the yukawas is small
compared to M1 or ,/, since in this limit the angle is unphysical and can be rotated
162
away by a redefinition of the higgsino fields. However we can see in Figure 6-2 that
for large yukawas the region above the diagonal M1 =
changes to a mixed state,
rather than being pure higgsino as before. Starting again with the large M1 limit
and decreasing M1 decreases the mass suppression of the t-channel bino exchange
diagram like in the 0 = 0 case, but the LSP also starts to mix more with the bino, an
effect that acts in the opposite direction and decreases ()eff.
This effect happens to
outweigh the former, forcing the LSP to shift to lower masses in order to annihilate
enough.
With 0 = 7r and yukawas large enough, there is an additional allowed region
for , < Mw. In this region the higgsino LSP is too light to annihilate to on-shell
gauge bosons, so the dominant annihilation channels are phase-space suppressed.
Furthermore if the splitting between the chargino and the LSP is large enough, the
effect of coannihilation with the chargino into photon and on-shell W is Boltzmann
suppressed, substantially decreasing the effective cross-section, and giving the right
relic abundance even with such a light higgsino LSP. Although acceptable from a
cosmological standpoint, this region is excluded by direct searches since it corresponds
to a chargino that is too light.
6.3.2
Bino Dark Matter
The region below the diagonal M1 = ,pcorresponds to a bino-like LSP. Recall that in
the absence of yukawa couplings pure binos in this model do not couple to anything
and hence cannot annihilate at all. Turning on the yukawas allows them to mix with
higgsinos which have access to gauge annihilation channels. For Au =
Ad =
0.1 this
effect is only large enough when M1 and p are comparable (in fact when they are
equal, the neutralino states are maximally mixed for arbitrarily small off-diagonal
terms), explaining the stripe near the diagonal in Figure 6-1. Once pugets larger than
- 1 TeV even pure higgsinos are too heavy to annihilate efficiently; this means that
mixing is no longer sufficient to decrease the dark matter relic density to acceptable
values and the stripe ends.
Increasing the yukawas beyond a certain value (Au =
163
Ad
=
0.88, which is slightly
larger than their values in Split SUSY, is enough), makes t-channel annihilation to
higgses become large enough that a bino LSP does not need to mix at all in order to
have the correct annihilation cross-section. This gives rise to an allowed region which
is in the shape of a stripe, where for fixed Ml the correct annihilation cross-section
is achieved only for the small range of
that gives the right t-channel suppression.
As M1 increases the stripe converges towards the diagonal in order to compensate
for the increase in LSP mass by increasing the cross-section. Once the diagonal is
reached this channel cannot be enhanced any further, and there is no allowed region
for heavier LSPs. In addition the cross-section for annihilation through an s-channel
resonant higgs, even though CP suppressed (see Section 6.5 for details), becomes large
enough to allow even LSPs that are very pure bino to annihilate in this way. The
annihilation rate for this process is not very sensitive to the mixing, explaining the
- 80 GeV. This line ends when Aigrows to
apparent horizontal line at M1 = 21Mhiggs
the point where the mixing is too small.
As in the higgsino case, taking 0 = 7rchanges the shape of the contours of constant
gaugino fraction and spreads them out in the plane (see Figure 6-2), making mixing
with higgsinos relevant throughout the region. For small M1 , the allowed region
starts where the mixing term is small enough for the combination of gauge and higgs
channels not to cause over-annihilation. Increasing M1 again makes the region move
towards the diagonal, where the increase in LSP mass is countered by increasing the
cross-section for the gauge channel from mixing more.
For either yukawa very large (Au = 1.38, Ad = 0.1), annihilation to higgses via
t-channel higgsinos is so efficient that this process alone is sufficient to give bino-like
LSPs the correct abundance. As M1 increases the allowed region again moves towards
the diagonal in such a way as to keep the effective cross-section constant by decreasing
the higgsino mass suppression, thus compensating for the increase in LSP mass. As
we remarked earlier since Ad is effectively zero in this case, the angle 0 is unphysical
and can be rotated away by a redefinition of the higgsino fields.
164
6.4
Direct Detection
Dark matter is also detectable
through elastic scatterings
off ordinary matter.
The
direct detection cross-section for this process can be divided up into a spin-dependent
and a spin-independent
dominant.
part; we will concentrate
As before we restrict to B =
significantly for intermediate
The spin-independent
a and
1r,
on the former since it is usually
we expect the result not to change
values.
interaction
takes place through
higgs exchange,
VIa the
yukawa couplings which mix higgsinos and binos. Since the only X~X~ h term in our
model involves the product of the gaugino and higgsino fractions, the more mixed our
dark matter is the more visible it will be to direct detection experiments.
This effect
can be seen in Fig 6-3 below.
Current bound
-----
1e-42 ,
--------------------------------
IS """""
I
-
"
,,~
~
1e-45 ,:
N
\
6
Proposed bound from
next-generation experiments
_
- .:::::-,-
---------------------- ------- ---
-------
-:
'2
o
Q)
C3
='
>::
1e-48
•
•
><
b
0=0, >.,,=0.88, >',J=0.88
0=7r, >.,,=0.88, >',J=0.88
•
0=0, >.,,=1.38, >'rI=O.l
•
0=7r, >.,,=0.1, >',J=0.1
Ie-51
1e-540
500
1000
1500
2000
2500
3000
Mx (GeV)
Figure 6-3: Spin-independent part of dark matter direct detection cross-section. The cu~rent
bound represents the CDMS limit [11], and, as an indicative value for the proposed bound from next
generation experiments, we take the projected sensitivity of SuperCDMS phase B [18].
Although it seems like we cannot currently use this measure as a constraint,
proportion
of our parameter
space will be accessible at next-generation
Since higgsino LSPs are generally more pure than bino-type
165
the major
experiments.
ones, the former will
escape detection as long as there is an order 100 keV splitting between its two neutral
components. This is is necessary in order to avoid the limit from spin-independent
direct detection measurements [12].
Also visible in the graph are the interesting discontinuities mentioned in [13],
corresponding to the opening up of new annihilation channels at MLSP = 1/2Mhiggs
through an s-channel higgs. We also notice a similar discontinuity at the top threshold
from annihilation to tt; this effect becomes more pronounced as the new yukawa
couplings increase.
6.5
Electric Dipole Moment
Since our model does not contain any sleptons it induces an electron EDM only at
two loops, proportional to sin(O) for 0 as defined above. This is a two-loop effect, we
therefore expect it to be close to the experimental bound for 0(1) 0. The dominant
diagram responsible for the EDM is generated by charginos and neutralinos in a loop
and can be seen in Figure 6-4 below. This diagram is also present in Split SUSY
where it gives a comparable contribution to the one with only charginos in the loop
[19, 20].
f'
f
Figure 6-4: The 2-loop contribution to the EDM of a fermion f.
The induced EDM is (see [19]):
dw
fd _
e
where
.
a2mf
1 Im ((OLOR*)5
0 L 0 R*)g (r, r)
24
f 2i 3mx/
mx'l~
2
8wr shMe
166
(6.2)
G(r° r)
=
dzI
idyY
l diy I/
-
ld
dy
yz(y+z/2)
yZ(K + K)
+ ((Ky)(zyK3
[ (y - 3Ki)y
2y)
n Ki]
3
4y(Ki+- 2(Ki
y) 2 + y)y + Ki(Ki
2(Ki -- y)
Y
and
17= _r- +r-,
oR = VNiexp
r-
2
r
mx
_-
,oi
OL=N3i
NTMNN = diag(mxl, mx 2 , mx) with real and positive diagonal elements. The sign
on the right-hand side of equation (6.2) corresponds to the fermion f with weak
isospin i
and f' is its electroweakpartner.
In principle it should be possible to cross-correlate the region of our parameter
space which is consistent with relic abundance measurements, with that consistent
with electron EDM measurements in order to further constrain our parameters. However since the current release of DarkSUSY does not support CP violating phases and
a version including CP violations seems almost ready for public release4 we leave an
accurate study of the consequences of non-zero CP phase in relic abundance and
direct detection calculations for a future work. We can still draw some interesting
conclusions by estimating the effect of non-zero CP phase. Because there is no reason
for these new contributions to be suppressed with respect to the CP-conserving ones
(for
of 0(1)), we might naively expect their inclusion to enhance the annihilation
cross-section by around a factor of 2, increasing the acceptable LSP masses by
V2
for constant relic abundance. This is discussed in greater detail in [21] (and [22] for
direct detection) in which we see that this observation holds for most of the parameter space. We must note, however, that in particular small regions of the space the
enhancement to the annihilation cross-section and the suppression to the elastic cross
section can be much larger, justifying further investigation of this point in future
4
Private communication with one of the authors of DarkSUSY.
167
work. With this assumption in mind we see in Figure 6-5 that although the majority
of our allowed region is below the current experimental
limit of de < 1.7
at 95% C.L. [23], most of it will be accessible to next generation
10-27 e cm
X
EDM experiments.
These propose to improve the precision of the electron EDM measurement
by 4 orders
of magnitude in the next 5 years, and maybe even up to 8 orders of magnitude,
ing permitting
fund-
[24, 25, 26]. We also see in this figure that CP violation is enhanced
on the diagonal where the mixing is largest.
This is as expected since the yukawas
that govern the mixing are necessary for there to be any CP violating phase at all.
For the same reason, decoupling either particle sends the EDM to zero .
3000
1 x 10
27
•
0=0, Au=O.88, Ad=O.88
•
O=7r, Au=O.88, Ad=O.88
,
\
\
~.
1.7 X 10-27
"
2000
>
0
v
'--"
~
1000
....• 3
x 10-27
-.
1 X 10-28
500
1500
1000
J-1
2000
2500
3000
(GeV)
Figure 6-5: Electron edm contours for 0 = 7r /2. The excluded region is bounded by the black
contours. Note that CP violation was not included in the relic density calculation, and the dark
matter plot is simply intended to indicate the approxi~ate region of interest for dark matter.
6.6
Gauge Coupling Unification
In this section we study the running of gauge couplings in our model at two loops.
The addition of higgsinos largely improves unification as compared to the 8M case,
168
but their effect is still not large enough and the model predicts a value for a,(Mz)
around 9a lower than the experimental value of as(Mz) = 0.119i0.002 [27]. Moreover
the scale at which the couplings unify is very low, around 1014 GeV, making proton
decay occur much too quickly to embed in a simple GUT theory 5 . These problems
can be avoided by adding the Split SUSY particle content at higher energies, as in
[29], at the cost of losing minimality; instead we choose to solve this problem by
embedding our minimal model in an extra dimensional theory. This decision is well
motivated: even though normal 4D GUTs have had some successes, explaining the
quark-lepton mass relations for example, and charge quantization in the SM, there
are many reasons why these simple theories are not ideal. In spite of the fact that
the matter content of the SM falls naturally into representations of SU(5), there are
some components that seem particularly resistant to this. This is especially true of
the higgs sector, the unification of which gives rise to the doublet-triplet splitting
problem. Even in the matter sector, although b-- unification works reasonably well
the same cannot be said for unification of the first two generations. In other words,
it seems like gauge couplings want to unify while the matter content of the SM does
not, at least; not to the same extent. This dilemma is easily addressed in an extra
dimensional model with a GUT symmetry in the bulk, broken down to the SM gauge
group on a brane by boundary conditions [30] since we can now choose where we put
fields based on whether they unify consistently or not. Unified matter can be placed
in the bulk whereas non-unified matter can be placed on the GUT-breaking brane.
The low energy theory will then contain the zero modes of the 5D bulk fields as well
as the brane matter. While solving many of the problems of standard 4D GUTs these
extra dimensional theories have the drawback of having a large number of discrete
choices for the location of the matter fields, as we shall see later.
We will consider a model with one flat extra dimension compactified on a circle of
radius R, with orbifolds S1 /(Z 2 x Z2), whose action is obtained from the identifications
5It is possible to evade the constraint from proton decay by setting some of the relevant mixing
parameters to zero [28]. However we are not aware of any GUT model in which such an assertion is
justified by symmetry arguments.
169
Z2
:
- 2r R - y,
Z2' :y
-- 7r
,
y [0, 277r
(6.3)
where y is the coordinate of the fifth dimension. There are two fixed points under this
action, at (0, w7R)and (rR/2, 37rR/2), at which are located two branes. We impose
an SU(5) symmetry in the bulk and on the y = 0 brane; this symmetry is broken
down to the SM SU(3) x SU(2) x U(1) on the other brane by a suitable choice
of boundary conditions. All fields need to have definite transformation properties
under the orbifold action - we choose the action on the fundamental to be
±P
--
and on the adjoint, ±[P,A], for projection operators Pz = (+, +, +, +, +) and
Pz, = (+, +, +,-,-).
Aa for a =
This gives SM gauge fields and their corresponding KK towers
1, ..., 12} (+, +) parity; and the towers Ad for
=
(13,..., 24} (+,-)
parity, achieving the required symmetry-breaking pattern. By gauge invariance the
unphysical fifth component of the gauge field, which is eaten in unitary gauge, gets
opposite boundary conditions. 6 We still have the freedom to choose the location of
the matter fields. In this model SU(5)-symmetric matter fields in the bulk will get
split by the action of the Z' orbifold: the SM 5 for instance will either contain a
massless d or a massless 1, with the other component only having massive modes.
Matter fields in the bulk must therefore come in pairs with opposite eigenvalues under
the orbifold projections, so for each SM generation in the bulk we will need two copies
of 10 + 5. This provides us with a simple mechanism to forbid proton decay from Xand Y-exchange and also to split the color triplet higgs field from the doublet. To
summarize, unification of SM matter fields in complete multiplets of SU(5) cannot
be achieved in the bulk but on the SU(5) brane, while matter on the SM brane is not
unified into complete GUT representations.
6
From an effective field theory point of view an orbifold is not absolutely necessary, our theory
can simply be thought of as a theory with a compact extra dimension on an interval, with two branes
on the boundaries. Because of the presence of the boundaries we are free to impose either Dirichlet
or Neumann boundary conditions for the bulk field on each of the branes breaking the SU(5) to
the SM gauge group purely by choice of boundary conditions and similarly splitting the multiplets
accordingly. Our orbifold projection is therefore nothing more than a further restriction to the set
of all possible choices we can make.
170
6.6.1
Running and matching
We run the gauge couplings from the weak scale to the cutoff A by treating our
model as a succession of effective field theories (EFTs) characterized by the differing
particle content at different energies. The influence of the yukawa couplings between
the higgsinos and the singlet on the two-loop running is negligible, hence it is fine to
assume that the singlet is degenerate with the higgsinos so there is only one threshold
from Mtop to the compactification scale 1/R, at which we will need to match with
the full 5D theory.
The SU(5)-symmetric bulk gauge coupling g5 can be matched on to the low energy
couplings at the renormalization scale M via the equation
1
27rR
-(M)
- + Ai(M)+ Ai(MR)
(6.4)
The first term on the right represents a tree level contribution from the 5d kinetic
term, Ai are similar contributions from brane-localized kinetic terms and Ai encode
radiative contributions from KK modes. The latter come from renormalization of the
4D brane kinetic terms which run logarithmically as usual.
To understand this in more detail let us consider radiative corrections to a U(1)
gauge coupling in an extra dimension compactified on a circle with no orbifolds, due
to a 5D massless scalar field [32]. Since 1/g' has mass dimension 1, by dimensional
analysis we might expect corrections to it to go like A + m log A where m is some
mass parameter in the theory. The linearly divergent term is UV sensitive and can
be reabsorbed into the definition of g5 , whereas the log term cannot exist since there
is no mass parameter in the theory. Hence the 5D gauge coupling does not run, and
neither does the 4D gauge coupling. This can also be interpreted from a 4D point of
view, where the KK partners of the scalar cut off the divergences of the zero mode.
Since there is no distinction between the wavefunctions for even (cosine) and odd
(sine) KK modes in the absence of an orbifold, and we know that the sum of their
contributions must cancel the log divergence of the 4D massless scalar, each of these
must give a contribution equal to -1/2 times that of the 4D massless scalar.
171
When we impose a Z2 orbifold projection and add two 3-branes at the orbifold
fixed points, the scalar field must now transform as an eigenstate of this orbifold action
and can either be even ((+, +), with Neuman boudary conditions on the branes),
or odd ((-,-),
with Dirichlet boundary conditions on the branes). This restricts
us to a subset of the original modes and the cancellation of the log divergence no
longer works. Since this running can only be due to 4D gauge kinetic terms localized
on the branes, where the gauge coupling is dimensionless and can therefore receive
logarithmic corrections, locality implies that the contribution from a tower of states
on a particular brane can only be due to its boundary condition on that brane, with
the total running equal to the sum of the contributions on each brane. In fact, it is
only in the vicinity of the brane that imposing a particular boundary condition has
any effect. As argued above, a (+, +) tower (excluding the zero mode) and a (-, -)
tower must each give a total contribution equal to -1/2 times that of the zero mode,
which corresponds to a coefficient of -1/4
to the running of each brane-localized
kinetic term. Taking into account the contribution of the zero mode we can say that
a tower of modes with + boundary conditions on a brane contributes +1/4 times
the corresponding 4D coefficient, while a - boundary condition contributes -1/4
times the same quantity. This argument makes it explicit that the orbifold projection
can be seen as a prescription on the boundary conditions of the fields in the extra
dimension, which only affect the physics near each brane.
Adding another orbifold projection as we are doing in this case also allows for
towers with (+, -) and (-, +) boundary conditions which, from the above argument,
both give a contribution of ±1/4 :F 1/4 = 0. The contribution of the (+, +) and
(-, -) towers clearly remains unchanged.
Explicitly integrating out the KK modes at one loop at the compactification scale
allows us to verify this fact, and also compute the constant parts of the threshold
corrections, which are scheme-dependent. In DR 7 we obtain [32]:
7
We use this renormalization scheme even though our theory is non-supersymmetric
threshold corrections in this scheme contain no constant part [31].
172
since 4D
Ai = 1(MR)
((bs-21bq+8bF)Fe(MR)+ (
-
21bG
+ 8biF)
F )
(6.5)
with
Fe(iR) = I-11
+00
Z= -
log(r)-
dt (t- 1 + t-
log(MR),
1/ 2 )
Fo = - log(2)
(6.6)
+oo
(3(it) - 1) ~ 0.02, 03(it) =
en 2
n=--oo
where bi, 'G 'F (,GF)
are the Casimirs of the KK modes of real scalars (not including
goldstone bosons), massive vector bosons and Dirac fermions respectively with even
(odd) masses 2n/R ((2n + 1)/R). As explained above, the logarithmic part of the
above expression is equal to exactly -1/2 times the contribution of the same fields
in 4D [33, 34]. Since the compactification scale 1/R will always be relatively close to
the unification scale A (so our 5D theory remains perturbative), it will be sufficient
for us to use one loop matching in our two loop analysis as long as the matching is
done at a scale M close to the compactification scale.
As an aside, from equation (6.4) we can get:
d
dAi =
bi - b2M M
8wr
(6.7)
where bi is shorthand for the combination (bS - 21bg + 8bF)/12 and biMM are the
coefficients of the renormalization group equations below the compactification scale
(see Appendix 6.9 for details). It is clear from this equation that it is unnatural to
require Ai(1/R)
< 1/(87r 2 ). The most natural assumption Ai(A)
1/(8wr2 ) gives
a one-loop contribution comparable to the tree level term, implying the presence of
some strong dynamics in the brane gauge sector at the scale A. We know that the 5D
gauge theory becomes strong at the scale 247r3 /g52,so from naive dimensional analysis
(NDA) (see for example [35]) we find that it is quite natural for A to coincide with
the strong coupling scale for the bulk gauge group.
173
Running equation (6.7) to the compactification scale we obtain
Ai(l/R) =
i(A) +
bi - bM
8
log(AR)
(6.8)
For AR > 1 the unknown bare parameter is negligible compared to the log-enhanced
part, and can be ignored, leaving us with a calculable correction.
27rR/g2 we expect AR
Using
GUT =
87r2 /gUT - 100. Keeping this in mind, we shall check
whether unification is possible in our model with AR in the regime where the bare
brane gauge coupling is negligible. To this purpose we will impose the matching
equation (6.4) at the scale A assuming Ai(A) = 0; we will then check whether the
value of AR found justifies this approximation.
In order to develop some intuition for the direction that these thresholds go in, we
can analyze the one-loop expression (with one-loop thresholds) for the gauge couplings
at Mz:
aqI(M) = ''
+ 47rAi(AR)+ Aonv(AR) +
log
+
)
(6.9)
bsM are the SM beta function coefficients (see Appendix 6.9), /u is the scale of the
= (
higgsinos and singlet, AConV
3-,
2, 0) are conversion factors from MS, in which
the low-energy experimental values for the gauge couplings are defined, to DR [36].
Taking the linear combination (9/14)a-
1
- (23/14)a21 + a3 1 allows us to eliminate
the A dependence as well as all SU(5)-symmetric terms, leaving
1
a (M-,)
9/14
al(M)
23/14 + 4(AR) +
log
conv +)
2
O(
(6.10)
lo
where X = (9/14)X 1 - (23/14)X2 + X3 for any quantity X. Recall that the leading
threshold correction from the 5D GUT is proportional to log(AR). The low-energy
value of a3 is therefore changed by
174
Ja3(MZ) =
2
3 (Mz))
27r
log(AR)
(6.11)
We still have the freedom to choose the positions of the various matter fields. In
order to determine the best setup for gauge coupling unification we need to keep in
mind two facts: the first is that adding SU(5) multiplets in the bulk does not have
any effect on a3(Mz); and the second is that b contains only contributions from (+, +)
modes (in unitary gauge none of our bulk modes have (-,-)
our SU(5) bulk multiplets are split into (+, +) and (+,-)
boundary conditions;
modes).
As stated at the beginning of this section our 4D prediction for a 3 (Mz) is too
low. Since fermions have a larger effect on running than scalars, this problem is
most efficiently tackled by splitting up the fermion content of the SM into non-SU(5)
symmetric parts in order to make b as positive as possible. Examining the particular
linear combination that eliminated the dependence on A at one loop we find that one
or more SU(5)-incomplete colored multiplets are needed in the bulk, or equivalently
the weakly-interacting part of the same multiplet has to be on one of the branes. Since
matter in the bulk is naturally split by the orbifold projections, this just involves
separating the pair of multiplets whose zero modes make up one SM family.
With this in mind we find that for fixed AR and
, since separating different
numbers of SM generations allows us to vary the low energy value of a3 anywhere
from its experimental value to several as off, gauge coupling unification really does
work in this model for some fraction of all available configurations. Although this
may seem a little unsatisfactory from the point of view of predictivity, the situation
can be somewhat ameliorated by further refining our requirements. For example, we
can go some way towards explaining the hierarchy between the SM fermion masses
by placing the first generation in the bulk, the second generation split between the
bulk and a brane and the third generation entirely on a brane. This way, in addition
to breaking the approximate flavor symmetry in the fermion sector we also obtain
helpful factors of order 1/v'K_ between the masses of the different generations. The
location of the higgs does not have a very large effect on unification, the simplest
175
choice would be to put it, as well as the higgsinos and singlet, on the SU(5)-breaking
brane, where there is no need to introduce corresponding color triplet fields. This also
helps to explain the hierarchy. In this model, which can be seen on the left-hand side
SU(5)
SU(3)xSU(2)xU(1)
SU(5)
H
52
102
10,
10,
102
5,
I 5 I'0
10
5:
10,
irR/2
nR/2
(a) Higgs on the broken brane
(b) Higgs in the bulk and third generation
on the SU(5) brane
Figure 6-6: Matter content of the two orbifold GUT models we propose.
of Figure 6-6, we also need to put our third generation and split second generation
on the same brane in order for them to interact with the higgs.
On the right is another model which capitalizes on every shred of evidence we
have about GUT physics: we put the higgs in the bulk in this case (recall that the
orbifold naturally gives rise to doublet-triplet splitting) so that we can switch the
third generation to the SU(5)-preserving brane and obtain b-r unification (see Figure
6-7) without having analogous relationships for the other two generations8 . We also
need to flip the positions of the first and second generations if we want to keep the
suppression of the mass of the first generation with respect to the second.
The low-energy values for a3 as a function of p in these two models can be seen
in Figure 6-8 for different AR. Note that unification can be acheived in the regime
8
As explained in [37] the two yukawa couplings Ab and AT, run differently only below the compactification scale. Because of locality, the fields living on the SU(5) brane do not feel the SU(5)
breaking until energies below the compactification scale; hence if they are unified at some high
energy they keep being unified until this scale.
176
>.,,(Mz)
0.028
0.027
0.026
40'
interval
'
~=------,--
0.025
0.024 --
,
--------,---========
--------------
AR=1oo
AR=lO
4D
0.023
0.022
1000
2000
3000
p. (GeV)
4000
5000
Figure 6-7: Low energy prediction for Ab(Mz), as a function of the higgsino mass
J.L, for the model
with the biggs in the bulk, for AR = 10,100 and for the 4D model. 40' interval taken from [27).
where AR
» 1, justifying
our initial assumption that the brane kinetic terms could
be neglected. We see that although the dependence on J.L is very slight, small J.L seems
to be preferred. However we cannot use this observation to put a firm upper limit on
J.L
because of the uncertainties associated with ignoring the bare kinetic terms on the
branes.
l)3(Mz)
0.125
:erval
I
0.12
0.115
AR=lOO
0.11
AR=lO
0.105
--- 1000
0.095
2000
------
3000
4000
5000
l' (GeV)
...................................................................................................
4D
Figure 6-8: Low energy prediction for Q:3(Mz), as a function of the higgsino mass J.L, for the model
with the higgs on the brane (black line), and for the model with the higgs in the bulk (green line),
for AR = 10,100, and for the 4D case (dashed line). Some typical values of 1/ R are shown.
The second configuration, Figure 6-6(b), also gives proton decay through the mixing of the third generation with the first two. From our knowledge of the CKM
matrix we infer that all mixing matrices will be close to the unit matrix, proton
177
decay will therefore be suppressed by off-diagonal elements. To minimize this suppression it is best to have an anti-neutrino and a strange quark in the final state.
The proton decay rate for this process was computed in [38] and is proportional
((9412 )
to
(I -
n)
[
I(Rt)
23 (Rti)l 3 (Ld) 3 12
where Rd,,u and Ld are the rotation
matrices of the right-handed down-type and up-type quarks, and the left-handed
down-type quarks, which are unknown. We assume that the 2-3 and the 1-3 mixing
elements are 0.05 and 0.01 respectively, similar to the corresponding CKM matrix
elements, giving a proton lifetime of
-p(P - K+iP) _ 6.6 x 1038 years x
1014Ge
V
4 x 10 35 years
(6.12)
for 1/R = 1.6 x 1013GeV. This is above the current limit from Super-Kamiokande of
1.9 x 1033 years at 90% C.L. [39, 40], although there are multi-megaton experiments
in the planning stages that are expected to reach a sensitivity of up to 6 x 1034 years
[40] . Given our lack of information about the mixing matrices involved9 , we see that
there might be some possibility that proton decay in this model will be seen in the
not-too-distant future.
6.7
Conclusion
The identification of a TeV-scale weakly-interacting particle as a good dark matter
candidate, and the unification of the gauge couplings are usually taken as indications
of the presence of low-energy SUSY. However this might not necessarily be the case.
If we assume that the tuning of the higgs mass can be explained in some other
unnatural way, through environmental reasoning for instance, then new possibilities
open up for physics beyond the SM. In this chapter we studied the minimal model
consistent with current experimental limits, that has both a good thermal dark matter
candidate and gauge coupling unification.
9
To this end we added to the SM two
Experiments have only constrained the particular combination that appears in the SM as the
CKM matrix.
178
higgsino-like particles and a singlet, with a singlet majorana mass of < 100 TeV in
order to split the two neutralinos and so avoid direct detection constraints. Making
the singlet light allowed for a new region of dark matter with mixed states as heavy
as
2.2 TeV, well beyond the reach of the LHC and the generic expectation for
a weakly interacting particle. Nevertheless we do have some handles on this model:
firstly via the 2-loop induced electron EDM contribution which is just beyond present
limits for CP angle of order 1, and secondly by the spin-independent direct detection
cross section, both of which should be accessible at next-generation experiments.
Turning to gauge coupling unification we saw that this was much improved at two
loops by the presence of the higgsinos. A full 4D GUT model is nevertheless excluded
by the smallness of the GUT scale - 1014GeV, which induces too fast proton decay.
We embedded the model in a 5D orbifold GUT in which the threshold corrections
were calculable and pushed ca in the right direction for unification (for a suitable
matter configuration). It is very gratifying that such a model can help explain the
pattern in the fermion mass hierarchy, give b-- unification, and predict a rate for
proton decay that can be tested in the future.
6.8
Appendix A: The neutralino mass matrix
We have a 3x3 neutralino mass matrix in which the mixing terms (see equation (6.2))
are unrelated to gauge couplings and are limited only by the requirement of perturbativity to the cutoff. It is possible to get a feel for the behavior of this matrix by
finding the approximate eigenvalues and eigenvectors in the limit of equal and small
off-diagonal terms (Av << M1 q±-/). The approximate eigenvalues and eigenvectors are
shown in the table below:
M2
M12 + 4A 2 v2
Gaugino fraction
M
1
Ml,.4
1-2
L2
/~2
4 2 v2 +
2
2
A v
-p)2
(Ml
0
2
2 2
V
179
for cos/) =
If M1 <<
1.
the first eigenstate will be the LSP. In the opposite limit the compo-
sition of the LSP is dependent upon the sign of cos(9). For cos(9) positive the second
eigenstate, which is pure higgsino, is the LSP, while for cos(O) negative, the mixed
third eigenstate becomes the LSP.
We also see from the table that in the pure higgsino case, a splitting of order
100 keV (sufficient to evade the direct detection constraint) can be achieved with a
singlet mass lighter than 109 GeV, where the upper limit corresponds to 0(1) yukawa
couplings.
6.9
Appendix B: Two-Loop Beta Functions for Gauge
Couplings
The two-loop RGE for the gauge couplings in our minimal model is
l
d
(--21r)idt1
=
b M + (4/¢)2
[
3
4j7rB
jMa
- di
]
- d)(2
with P-function coefficients
104
18
5
44
6
5
14
12
11
9
2
25
bm
__
-7
BMM=
6
10
(17 3
d= 10,2,2);
5
-26
d(3
d=
20'4')
The running of the yukawa couplings is the same as in [9] but we will reproduce
their RGEs here for convenience - we ignore all except the top yukawa coupling (we
found that our two new yukawas do not have a significant effect).
(47r) 2 At = At -3 E
(~~~~~i=1lh[~~~ih+
47rcii
180
t ++
(AU + Ad)
withc= (17,3 )
The two-loop coupled RGEs can be solved analytically if we approximate the top
yukawa coupling as a constant over the entire range of integration (see [36]for a study
on the validity of this approximation). The solution is
- (M)
-
1
2M+
2r
2
14 BM
47
W
M
A
n(l+
b
47r I
181
G(A)
lnM -
3 2 73
diA21nA
t
M
Bibliography
[1] S. Perlmutter et al. [Supernova Cosmology Project Collaboration], "Measurements
of Omega and Lambda from 42 High-Redshift Supernovae," Astrophys. J. 517
(1999) 565, astro-ph/9812133.
[2] D. N. Spergel et al. [WMAP Collaboration], "First Year Wilkinson Microwave
Anisotropy Probe (WMAP) Observations: Determination of Cosmological Parameters," Astrophys. J. Suppl. 148 (2003) 175, astro-ph/0302209. C. L. Bennett
et al., "First Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Preliminary Maps and Basic Results," Astrophys. J. Suppl. 148, 1 (2003),
astro-ph/0302207.
[3] R. Barbieri and A. Strumia, "What is the limit on the Higgs mass?," Phys. Lett.
B 462, 144 (1999), hep-ph/9905281.
[4] L. Giusti, A. Romanino and A. Strumia, "Natural ranges of supersymmetric signals," Nucl. Phys. B 550 (1999) 3, hep-ph/9811386.
[5] S. Weinberg, "Anthropic Bound On The Cosmological Constant," Phys. Rev.
Lett. 59 (1987) 2607.
[6] V. Agrawal, S. M. Barr, J. F. Donoghue and D. Seckel, "The anthropic principle
and the mass scale of the standard model," Phys. Rev. D 57, 5480 (1998), hepph/9707380.
182
[7] N. Arkani-Hamed and S. Dimopoulos, "Supersymmetric unification without low
energy supersymmetry and signatures for fine-tuning at the LHC," JHEP 0506,
073 (2005) [arXiv:hep-th/0405159].
[8] N. Arkani-Hamed, S. Dimopoulos and S. Kachru, "Predictive landscapes and new
physics at a TeV," arXiv:hep-th/0501082.
[9] G. F. Giudice and A. Romanino, "Split supersymmetry," Nucl. Phys. B 699, 65
(2004) [Erratum-ibid.
B 706, 65 (2005)], hep-ph/0406088.
[10] M. W. Goodman and E. Witten, "Detectability Of Certain Dark-Matter Candidates," Phys. Rev. D 31, 3059 (1985).
[11] D. S. Akerib et al. [CDMS Collaboration], "Limits on spin-independent WIMP
nucleon interactions from the two-tower run of the Cryogenic Dark Matter
Search," arXiv:astro-ph/0509259.
[12] D. R. Smith and N. Weiner, "Inelastic dark matter," Phys. Rev. D 64, 043502
(2001), hep-ph/0101138.
[13] A. Pierce, "Dark matter in the finely tuned minimal supersymmetric standard
model," Phys. Rev. D 70 (2004) 075006, hep-ph/0406144.
[14] A. Masiero, S. Profumo and P. Ullio, "Neutralino dark matter detection in split
supersymmetry scenarios," Nucl. Phys. B 712, 86 (2005) [arXiv:hep-ph/0412058].
L. Senatore, "How heavy can the fermions in split SUSY be? A study on gravitino and extradimensional LSP," Phys. Rev. D 71, 103510 (2005) [arXiv:hepph/0412103].
[15] P. Gondolo, J. Edsjo, P. Ullio, L. Bergstrom, M. Schelke and E. A. Baltz, "Dark-
SUSY: A numerical package for supersymmetric dark matter calculations," ,astroph/0211238; J. Edsjo, M. Schelke, P. Ullio and P. Gondolo, "Accurate relic densities with neutralino, chargino and sfermion coannihilations in mSUGRA," JCAP
0304, 001 (2003), hep-ph/0301106]; P. Gondolo and G. Gelmini, "Cosmic Abundances Of Stable Particles: Improved Analysis," Nucl. Phys. B 360, 145 (1991).
183
[16] A. Provenza, M. Quiros and P. Ullio, "Electroweak baryogenesis, large Yukawas
and dark matter," JHEP 0510, 048 (2005) [arXiv:hep-ph/0507325].
[17] J. Edsjo and P. Gondolo, "Neutralino relic density including coannihilations,"
Phys. Rev. D 56, 1879 (1997), hep-ph/9704361];
[18] P.
II
L. Brink
dark
matter
et
al.
[CDMS-II
search:
Collaboration],
SuperCDMS,"
"Beyond the
eConf C041213,
CDMS-
2529
(2004)
[arXiv:astro-ph/0503583]. Useful comparison can be done using the tools of
http://dendera.berkeley.edu/plotter/entryform.html
[19] D. Chang, W. F. Chang and W. Y. Keung, "Electric dipole moment in the split
supersymmetry models," Phys. Rev. D 71 (2005) 076006, hep-ph/0503055.
[20] N. Arkani-Hamed, S. Dimopoulos, G. F. Giudice and A. Romanino, "Aspects of
split supersymmetry," Nucl. Phys. B 709, 3 (2005) [arXiv:hep-ph/0409232].
[21] P. Gondolo and K. Freese, "CP-violating effects in neutralino scattering and
annihilation," JHEP 0207, 052 (2002), hep-ph/9908390.
[22] T. Falk, A. Ferstl and K. A. Olive, "Variations of the neutralino elastic crosssection with CP violating phases," Astropart. Phys. 13, 301 (2000) [arXiv:hepph/9908311].
[23] B. C. Regan, E. D. Commins, C. J. Schmidt and D. DeMille, "New limit on the
electron electric dipole moment," Phys. Rev. Lett. 88 (2002) 071805.
[24] D. Kawall, F. Bay, S. Bickman, Y. Jiang and D. DeMille, "Progress towards
measuring the electric dipole moment of the electron in metastable PbO," AIP
Conf. Proc. 698 (2004) 192.
[25] S. K. Lamoreaux, "Solid state systems for electron electric dipole moment and
other fundamental measurements," nucl-ex/0109014.
[26] Y. K. Semertzidis,
"Electric dipole moments of fundamental
arXiv:hep-ex/0401016.
184
particles,"
[27] S. Eidelman et al. [Particle Data Group], "Review of particle physics," Phys.
Lett. B 592 (2004) 1.
[28] I. Dorsner and P. F. Perez, "How long could we live?," Phys. Lett. B 625, 88
(2005) [arXiv:hep-ph/0410198].
[29] I. Antoniadis, A. Delgado, K. Benakli, M. Quiros and M. Tuckmantel, "Splitting
extended supersymmetry," arXiv:hep-ph/0507192.
[30] Y. Kawamura, "Gauge symmetry reduction from the extra space S(1)/Z(2),"
Prog. Theor. Phys. 103, 613 (2000), hep-ph/9902423. A. Hebecker and J. MarchRussell, "A minimal S(1)/(Z(2) x Z'(2)) orbifold GUT," Nucl. Phys. B 613, 3
(2001), hep-ph/0106166. L. J. Hall and Y. Nomura, "Gauge coupling unification
from unified theories in higher dimensions," Phys. Rev. D 65, 125012 (2002),
hep-ph/0111068. G. Altarelli and F. Feruglio, "SU(5) grand unification in extra
dimensions and proton decay," Phys. Lett. B 511, 257 (2001), hep-ph/0102301.
L. J. Hall and Y. Nomura, "Grand unification in higher dimensions," Annals
Phys. 306, 132 (2003), hep-ph/0212134. P. C. Schuster, "Grand unification in
higher dimensions with split supersymmetry," hep-ph/0412263. A. B. Kobakhidze,
"Proton stability in TeV-scale GUTs," Phys. Lett. B 514, 131 (2001) [arXiv:hepph/0102323].
[31] I. Antoniadis, C. Kounnas and K. Tamvakis, "Simple Treatment Of Threshold
Effects," Phys. Lett. B 119, 377 (1982).
[32] R. Contino, L. Pilo, R. Rattazzi and E. Trincherini, "Running and matching
from 5 to 4 dimensions," Nucl. Phys. B 622, 227 (2002), hep-ph/0108102.
[33] S. Weinberg, "Effective Gauge Theories," Phys. Lett. B 91, 51 (1980).
[34] L. J. Hall, "Grand Unification Of Effective Gauge Theories," Nucl. Phys. B 178,
75 (1981).
185
[35] Z. Chacko, M. A. Luty and E. Ponton, "Massive higher-dimensional gauge
fields as messengers of supersymmetry breaking," JHEP 0007, 036 (2000), hepph/9909248.
[36] P. Langacker and N. Polonsky, "Uncertainties in coupling constant unification,"
Phys. Rev. D 47, 4028 (1993), hep-ph/9210235.
[37] L. J. Hall and Y. Nomura, "A complete theory of grand unification in five dimensions," Phys. Rev. D 66, 075004 (2002), hep-ph/0205067.
[38] A. Hebecker and J. March-Russell, "Proton decay signatures of orbifold GUTs,"
Phys. Lett. B 539, 119 (2002), hep-ph/0204037.
[39] Y. Hayato et al. [Super-Kamiokande Collaboration], "Search for proton decay
through p - anti-nu K+ in a large water Cherenkov detector," Phys. Rev. Lett.
83 (1999) 1529, hep-ex/9904020.
[40] Y. Suzuki et al. [TITAND Working Group], "Multi-Megaton water Cherenkov
detector for a proton decay search: TITAND (former name: TITANIC)," hepex/0110005.
186
Chapter 7
Conclusions
I have reported on some of the research work I have done during my years as a
graduate student.
My research interest has focused mainly on the connections between Particle
Physics and Cosmology, of which I think I have been able to explore many of the
different aspects.
Thanks to the very recent improvement in the experimental techniques in Observational Cosmology, the field of Cosmology has become a field where new observations
can motivate new ideas from Theoretical Physics, as for example with the observation
of Dark Matter, of the tyne anisotropy of the CMB, and recently of the acceleration
of the universe. Viceversa it has become also a field where new ideas from Theoretical
Physics can be applied to the early universe with the purpose either of testing them
against some phases that we consider well understood, or of gaining through them
some insight on phases less understood, as for example is the case of the inflationary
phase.
During my research work, I think I have been able to deepen most of these aspects.
Concerning that part which deals with using new high energy theories to develop
new possible models for the early universe, in chapter 2 I have shown a model for
the early universe where a new kind of inflation is developed. This new inflationary
model is based on the condensate of a ghost field, which is a new mechanism for
obtaining a stable Lorentz violating state in quantum field theory which has been
187
understood only very recently. When applied to the inflationary phase, the ghost
condensate predicts some new features, such as a large amount of non-Gaussianities,
which, because of the large information this kind of signal carries on, if detected, it
would both tell us with great confidence that inflation did occur in the past, and
also that it occurred in a way very similar to the one of ghost condensation. This
inflationary model is also the first one in which it is shown that the Hubble rate H
can actually grow with time.
The work on non-Gaussianities has found a natural development in the direct
analysis of the data from the most advanced experiment on the CMB: the WMAP
experiment, as I have shown in chapter 5. Experimental data are at the origin of
Physics, and it is really emotional to be able to deal with them. The work on the
analysis of the non-Gaussian signal in the WMAP data originates from a long theoretical work during which the generation of a non-Gaussian inflationary signal in the
CMB and its importance as a possible smoking gun for inflation was progressively
understood.
In the very last few years, several inflationary models have emerged
which are distinguishable from the standard slow roll scenario exactly for the predicted amount and characteristics of the non-Gaussian signal. After this theoretical
work, my collaborators and I realized that in the WMAP data there was more information on the non-Gaussian signal than what had already been extracted.
We
therefore decided to develop a method to better exploit the experimental data, and
we directly did the analysis. No evidence of a non-Gaussian signal was found, but we
improved and completed the available limits, so that the limits we give are at present
the best currently available.
This last work represents to me that part of the field of Cosmology in which new
theoretical ideas begin to be tested by new cosmological observations. In this thesis
I have shown some other works in this direction, in which some other aspects of the
history of the universe that we consider well understood are used to constraint models
of Physics beyond the Standard Model.
In chapters 3,4 and 6, I have shown how some observational constraints coming
from Dark Matter relic abundance, Baryogenesis, or Big Bang Nucleosynthesis, can
188
be applied to test ideas motivated by the Landscape.
The Landscape itself, istead, represent the most recent important idea in Theoretical Physics which is, at least partially, motivated by Cosmological Observations.
In fact, the Landscape is a scenario which has recently emerged from String Theory
in which the UV theory which describes our universe has a huge number of different
low energy configurations, in which the parameters that we are able to observe scan
from vacuum to vacuum. Together with the cosmological observation that the cosmological constant in our universe is not-null, and that it is roughly comparable to
the upper limit it has in order for structures to be present in our universe, these theoretical considerations have led to the proposal that the reason why the cosmological
constant (as well as the higgs mass) are small is not dynamical, as usually thought,
but rather environmental, where by environmental we mean that their value should
be such that the environment in our universe is such that physical observers can form
in it. This has led to the proposal of several specific models, some of which I have
then applied to the early universe and studied their constraint.
In chapter 3, I have considered the case of Split Susy, the first of these theories,
where it is proposed that the supersymmetric partners of the Standard Model are split
into scalar, which are very heavy at an intermediate scale between the weak scale and
the GUT scale, and the fermions, which are instead at the weak scale protected by
chiral symmetry. The lightness of the higgs is then not explained dynamically, but
it is tuned to be light for environmental reasons. In this model it is the fact that
the lightest supersymmetric partner (LSP) should be a thermal relic and made up
the dark matter of the universe that fixes the fermions at around the weak scale,
with deep consequences for a possible detection at LHC. This would in fact not be
true if the gravitino or another dark sector particle were the LSP, and I have studied
in detail the dark matter relic abundance in this case, and its consequences on the
spectrum of the weakly interacting fermionic partners.
In the same direction, in chapter 4, I have studied a model which explains the
tuning of the higgs through a mechanism in which baryons, which are necessary
for in our universe there to be life, are present in our universe only if the higgs is
189
parametrically light. Baryons are produced at the electroweak phase transition in the
so called electroweak baryogenesis scenario, and I have studied in detail the predicted
baryon abundance.
Taking this into account, the Physics associated with the CP
violation then becomes very interesting when it is realized that the natural predicted
electron electric dipole moment is parametrically just beyond current experimental
limits, at a level reachable by next generation experiments.
Eventually, in chapter 6, I have illustrated a study of the minimal model beyond
the standard model which accounts for the dark matter in the universe and gauge
coupling unification, where again the higgs is assumed to be light for environmental
reasons. We have realized that the right amount of dark matter abundance can be
achieved with particles of unusual large mass, while gauge coupling unification works
extremely well, as we have explicitly shown embedding the model in a five dimensional
theory.
I think that thanks to these works I have been able to gain some good understanding of the field of Cosmology and Particle Physics. It is then very exciting to
realize that in the very next few years, with the turning on of new experiments in
Cosmology, such as the Planck satellite, and in Particle Physics as LHC, many of the
ideas I have worked on during my studies for a PhD will be possibly tested, and our
understanding of both the fields potentially revolutionized. A detection in the CMB
of a tilt in the power spectrum of the scalar perturbations, or of a polarization induced
by a background of gravitational waves, or of a non-Gaussian signal would shed light
into the inflationary phase of the universe, and therefore also on the physics at very
high energies. Combined with a definite improvement in the data on the Supernovae,
expected in the very near future, these new experiments might shed light also on the
great mystery of the present value of the Dark Energy, ruling out or giving further
support to the idea of the Landscape. Similar in this will be the turning on of LHC
which will probably help us understand if the solution to the hierarchy problem is to
be solved in some natural way, as for example with TeV scale Supersymmetry, or if
there is evidence of a tuning which points towards the presence of a Landscape.
190